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\title{\bf Evaluations of Hecke algebra traces at Kazhdan-Lusztig basis elements}  

% input author, affilliation, address and support information as follows;
% the address should include the country, and does not have to include
% the street address 

\author{Samuel Clearman\\
\small Department of Mathematics\\[-0.8ex]
\small Lehigh University\\[-0.8ex] 
\small Bethlehem, PA, U.S.A.\\
\small\tt sam@samclearman.com\\
\and
Matthew Hyatt\\
\small Department of Mathematics\\[-0.8ex]
\small Pace University\\[-0.8ex] 
\small Pleasantville, NY, U.S.A.\\
\small\tt mhyatt@pace.edu\\
\and
Brittany Shelton\\
\small Department of Mathematics\\[-0.8ex]
\small Albright College\\[-0.8ex] 
\small Reading, PA, U.S.A.\\
\small\tt bshelton@albright.edu\\
\and
Mark Skandera\\
\small Department of Mathematics\\[-0.8ex]
\small Lehigh University\\[-0.8ex] 
\small Bethlehem, PA, U.S.A.\\
\small\tt mas906@lehigh.edu
}

\date{\dateline{Feb 3, 2015}{Mar 28, 2016}{Apr 15, 2016}\\
\small Mathematics Subject Classifications: 05E10, 20C08, 05E05}

\begin{document}

\maketitle

\begin{abstract}
%\vspace{.15 in}
For irreducible characters 
$\{ \chi_q^\lambda \,|\, \lambda \vdash n \}$,
induced sign characters
$\{ \epsilon_q^\lambda \,|\, \lambda \vdash n \}$,
and induced trivial characters
$\{ \eta_q^\lambda \,|\, \lambda \vdash n \}$
of the Hecke algebra $\hnq$, 
and Kazhdan-Lusztig basis elements
$C'_w(q)$ with $w$ \avoidingp,
we combinatorially interpret the polynomials
$\smash{\chi_q^\lambda(q^{\ell(w)/2} C'_w(q))}$,
$\smash{\epsilon_q^\lambda(q^{\ell(w)/2} C'_w(q))}$,
and
$\smash{\eta_q^\lambda(q^{\ell(w)/2}  C'_w(q))}$.
%$\smash{\chi_q^\lambda(\qp{\ell(w)}2 C'_w(q))}$,
%$\smash{\epsilon_q^\lambda(\qp{\ell(w)}2 C'_w(q))}$,
%and
%$\smash{\eta_q^\lambda(\qp{\ell(w)}2 C'_w(q))}$.
This provides a new algebraic interpretation of 
chromatic quasisymmetric functions of Shareshian and Wachs,
and a new combinatorial interpretation of special cases of results of 
Haiman.
We prove similar results 
for other $\hnq$-traces,
and confirm
a formula conjectured by Haiman.

  \bigskip\noindent \textbf{Keywords:} Hecke algebra, character, trace, 
Kazhdan-Lusztig basis, chromatic quasisymmetric function, planar network, 
$P$-tableau, pattern avoidance
\end{abstract}




\section{Introduction}\label{s:intro}

The {\em symmetric group algebra} $\zsn$
%generated over $\mathbb C$ by $s_1,\dotsc, s_{n-1}$, 
and 
the 
{\em (Iwahori-) Hecke algebra} $\hnq$
have similar presentations as algebras over $\mathbb Z$ and $\zqq$
respectively, with 
multiplicative identity elements $e$ and $T_e$,
generators 
$s_1,\dotsc, s_{n-1}$ and 
%generated over $\cqq$ by 
$T_{s_1},\dotsc, T_{s_{n-1}}$, 
%are presented by similar 
and relations
\begin{equation*}
\begin{alignedat}{3}
s_i^2 &= e &\qquad
T_{s_i}^2 &= (q-1) T_{s_i} + qT_e &\qquad
&\text{for $i = 1, \dotsc, n-1$},\\
s_is_js_i &= s_js_is_j &\qquad
T_{s_i}T_{s_j}T_{s_i} &= T_{s_j}T_{s_i}T_{s_j} &\qquad
&\text{for $|i - j| = 1$},\\
s_is_j &= s_js_i &\qquad
T_{s_i}T_{s_j} &= T_{s_j}T_{s_i} &\qquad
&\text{for $|i - j| \geq 2$}.
\end{alignedat}
\end{equation*}
%where $e$ and $\wT_e$ are the respective multiplicative identity elements.
Analogous to the {\em natural basis} $\sn$ of $\zsn$ 
%$\{ w \,|\, w \in \sn \}$ 
is the {\em natural basis} $\{ T_w \,|\, w \in \sn \}$
of $\hnq$,
where we define
$T_w = T_{s_{i_1}} \cdots T_{s_{i_\ell}}$ whenever $s_{i_1} \cdots s_{i_\ell}$
is a reduced (short as possible) expression for $w$ in $\sn$.  
We call $\ell$ the {\em length}
of $w$ and write $\ell = \ell(w)$.  It is known that $\ell(w)$ is equal
to $\inv(w)$, the number of {\em inversions} 
in the one-line notation $w_1 \cdots w_n$ of $w$, i.e., the number of pairs
$(i,j)$ with $i < j$ and $w_i > w_j$.
%(See, e.g., \cite{BjBrentiBruhat}, \cite{Hum} for defintions.) 
%The ``longest'' element of $\sn$, written $w_0$, has one-line notation
%$n \cdots 1$ and length $n(n-1)/2$. 
The specialization of $\hnq$ at $\qp12 = 1$ is isomorphic to $\zsn$.

In addition to the natural bases of $\zsn$ and $\hnq$, we have 
the (signless) {\em Kazhdan-Lusztig bases}~\cite{KLRepCH} 
$\{ C'_w(1) \,|\, w \in \sn \}$,
$\{ C'_w(q) \,|\, w \in \sn \}$, defined in terms of certain 
{\em Kazhdan-Lusztig polynomials} $\{ P_{v,w}(q) \,|\, v,w \in \sn \}$ 
in $\mathbb N[q]$ by
\begin{equation}\label{eq:KLbasis}
C'_w(1) = \sum_{v \leq w} P_{v,w}(1) v, 
\qquad
%C'_w(q) = \qiew \sum_{v \leq w} P_{v,w}(q) T_v,
C'_w(q) = \qiew \sum_{v \leq w} P_{v,w}(q) T_v,
\end{equation}
where $\leq$ denotes the Bruhat order on $\sn$
and we define 
$\qvw = \qp{\ell(w) - \ell(v)}2$.
%$\qew = \qp{\ell(w)}2$.
%(See \cite{Hum}.)
%(See, e.g., \cite{BjBrentiBruhat}.) 
%(See \cite{KLRepCH}.)
We have the identity $P_{v,w}(q) = 1$ when $w$ \avoidsp,
i.e., when no subword $w_{i_1} w_{i_2} w_{i_3} w_{i_4}$ of $w_1 \cdots w_n$
consists of letters which appear in the same relative order as 
$3412$ or $4231$~\cite{LakSan}.
%satisfies $w_{i_3} < w_{i_4} < w_{i_1} < w_{i_2}$
%or $w_{i_4} < w_{i_2} < w_{i_3} < w_{i_1}$.  
These particular permutations are of interest to algebraic geometers
because they correspond to smooth Schubert varieties.
(See \cite[Ch.\,4, Ch.\,13]{BilleyLak}.)
%($\star$ Say something about this pattern avoidance.)


%($\star$ Need more info on the symmetric group here or elsewhere?
%Young subgroups, 
%cycle type, one-line notation, 
%left descent, Bruhat order,
%dominance, partition, transpose, 
%pattern avoidance, 
%etc.)

Representations of $\zsn$ and $\hnq$ are often studied in terms of 
%$\mathbb C$- and $\cqq$-
linear maps called 
{\em characters}.
(See \cite[Ch.\,1]{Sag} for definitions.)
%(See, e.g., \cite{Sag}.)
%Like other algebras, $\zsn$ and $\hnq$ may be studied by 
%representing their elements as linear transformations on a 
%space called an {\em $\sn$-module} or {\em $\hnq$-module}.
%The trace $\chi$ of each such linear transformation is called 
%an $\sn$- or $\hnq$-{\em character}, 
%and has the {\em class function} property 
%$\chi(\sigma \tau \sigma^{-1}) = \chi(\tau)$ for $\sigma$ invertible,
%and the {\em trace} property 
%$\chi(\sigma \tau) = \chi(\tau \sigma)$ in general.
The 
%$\mathbb C$-span 
span
of the $\sn$-characters is called the space of
{\em $\sn$-class functions}, and has
dimension equal to the number of integer partitions of $n$.  
%(See \cite{Sag}
%, \cite{StanEC1}, \cite{StanEC2} for definitions.)  
%Letting $\lambda \vdash n$ denote that $\lambda$ is a partition of $n$,
%we 
Three well-studied bases 
%whose elements are indexed by partitions $\lambda$ of $n$
are the irreducible characters $\{ \chi^\lambda \,|\, \lambda \vdash n \}$,
induced sign characters $\{ \epsilon^\lambda \,|\, \lambda \vdash n \}$, and
induced trivial characters $\{ \eta^\lambda \,|\, \lambda \vdash n \}$,
where $\lambda \vdash n$ denotes that $\lambda$ is a partition of $n$.
Letting $\slambda$ denote the Young subgroup of $\sn$ of type $\lambda$,
we have
\begin{equation*}
\epsilon^\lambda \defeq \mathrm{sgn}\ntnsp\uparrow_{\slambda}^{\sn}, \qquad
\eta^\lambda \defeq \mathrm{triv}\ntnsp\uparrow_{\slambda}^{\sn}\ntnsp.
\end{equation*}
%(See \cite[Ch.\,1]{Sag} for definitions.)
The 
%$\cqq$-span 
span 
of the $\hnq$-characters, called the space of
{\em $\hnq$-traces}, has the same dimension and analogous character bases
% and has
%dimension is equal to the number of integer partitions of $n$.  
%($\star$ See appropriate ref.)  
%Letting $\lambda \vdash n$ denote that $\lambda$ is a partition of $n$,
%we 
%($\star$ Let $\lambda \vdash n$ denote that $\lambda$ is a partition of $n$.)
%Three well-studied bases 
%whose elements are indexed by partitions $\lambda$ of $n$
%are the 
%irreducible characters $\{ \chi_q^\lambda \,|\, \lambda \vdash n \}$,
%induced sign characters $\{ \epsilon_q^\lambda \,|\, \lambda \vdash n \}$, and
%induced trivial characters $\{ \eta_q^\lambda \,|\, \lambda \vdash n \}$,
$\{ \chi_q^\lambda \,|\, \lambda \vdash n \}$,
$\{ \epsilon_q^\lambda \,|\, \lambda \vdash n \}$, 
$\{ \eta_q^\lambda \,|\, \lambda \vdash n \}$,
specializing at $\smash{\qp12} = 1$ to the $\sn$-character bases.
Each of the two spaces has a fourth basis consisting of 
{\em monomial} class functions $\{ \phi^\lambda \,|\, \lambda \vdash n \}$ 
or traces $\{ \phi_q^\lambda \,|\, \lambda \vdash n \}$,
and a fifth basis consisting of
%consists of 
{\em power sum} class functions $\{ \psi^\lambda \,|\, \lambda \vdash n \}$
or traces $\{ \psi_q^\lambda \,|\, \lambda \vdash n \}$.
These are defined via the inverse Kostka numbers 
$\{ K_{\lambda,\mu}^{-1} \,|\, \lambda, \mu \vdash n\}$ 
and the numbers
$\{ L_{\lambda,\mu} \,|\, \lambda, \mu \vdash n\}$ 
of row-constant Young tableaux of shape $\lambda$ and content $\mu$ by
%(See, e.g., \cite{RemTrans}.)
%by
\begin{equation}\label{eq:moncfdef}
\phi^\lambda \defeq \sum_\mu K_{\lambda,\mu}^{-1} \chi^\mu,
\quad \phi_q^\lambda \defeq \sum_\mu K_{\lambda,\mu}^{-1} \chi_q^\mu,
\qquad \psi^\lambda \defeq \sum_\mu L_{\lambda,\mu} \phi^\mu,
\quad \psi_q^\lambda \defeq \sum_\mu L_{\lambda,\mu} \phi_q^\mu.
\end{equation}
%where $\{ K_{\lambda,\mu}^{-1} \}$ 
%are the {\em inverse Kostka numbers}.
%\vspace{-1mm}
Each of these functions is not a character, but is a 
% which are not characters, are named
%We have named of these functions, each of which is a 
difference of two characters.
In each space, 
the five bases are related to one another by the same transition matrices
which relate the
%standard bases of the space $\lambda_n$
Schur $\{ s_\lambda \,|\, \lambda \vdash n \}$, 
elementary $\{ e_\lambda \,|\, \lambda \vdash n \}$,  
complete homogeneous $\{ h_\lambda \,|\, \lambda \vdash n \}$, 
monomial $\{ m_\lambda \,|\, \lambda \vdash n \}$, 
and power sum $\{ p_\lambda \,|\, \lambda \vdash n \}$ 
bases of the space $\Lambda_n$
of homogeneous degree $n$ symmetric functions.
It follows from the theory of symmetric functions
that basis elements in each space are integer linear combinations
of irreducible characters in that space. 
(For more information on 
the transition matrices and symmetric functions, 
see \cite[Sec.\,2]{RemTrans}, \cite[Ch.\,7]{StanEC2}, respectively.)
A correspondence between these class functions 
and symmetric functions
%, are inspired by 
is given by the Frobenius 
{\em characteristic map} 
%from the space of $\sn$-class functions
%to the space $\Lambda_n$ of homogeneous degree $n$ symmetric functions,
%associates a symmetric function
\begin{equation*}
\mathrm{ch}(\theta) \defeq \frac1{n!} \sum_{w \in \sn} \theta(w) p_{\mathrm{ctype}(w)}, 
\end{equation*}
where $\mathrm{ctype}(w)$ is the cycle type of $w$.
%(See 
%\cite{GPCharHecke}, 
%\cite{RemTrans},
%\cite{HaimanHecke}, 
%\cite{StanEC2}, 
%\cite{StemConj}.)
In particular, we have 
$\mathrm{ch} (\chi^\lambda) = s_\lambda$,
$\mathrm{ch} (\epsilon^\lambda) = e_\lambda$,
$\mathrm{ch} (\eta^\lambda) = h_\lambda$,
$\mathrm{ch} (\phi^\lambda) = m_\lambda$,
$\mathrm{ch} (\psi^\lambda) = p_\lambda$,
which explains our names for the fourth and fifth bases.
%\cite{GPCharHecke}, 
%\cite{HaimanHecke}, 
%\cite{StanEC2}, 
%\cite{StemConj}.)
We naturally use
the Greek ancestors $\epsilon$, $\eta$ of $e$, $h$
in our class function notation, but we prefer not to use
the ancestors
%for 
%Rather than using the Greek letters 
$\sigma$, $\mu$, $\pi$ of $s$, $m$, $p$.
%for the class functions and traces which correspond 
%to Schur, monomial, and power sum symmetric functions.
Instead we follow the standard practices of using $\chi$ for 
irreducible characters, while reserving $\mu$ for integer
partitions, and $\pi$ for path families in planar networks.
We follow \cite{HaimanHecke}, \cite{StemConj} in using
$\phi$,
% following 
%notation in 
and 
%our choic in using $\phi$ for monomial
%class functions and traces, and we use 
we use $\psi$ 
%for power sum class functions and traces simply 
because {\em psi} begins with $p$. 



%A fifth basis of each space consists of {\em power sum} class functions
%$\{ \psi^\lambda \,|\, \lambda \vdash n \}$
%or traces $\{ \psi_q^\lambda \,|\, \lambda \vdash n \}$
%\begin{equation}\label{eq:powcfdef}
%\psi^\lambda \defeq \sum_\mu L_{\lambda,\mu} \phi^\mu,
%\qquad \psi_q^\lambda \defeq \sum_\mu L_{\lambda,\mu} \phi_q^\mu,
%\end{equation}
%where $L_{\lambda,\mu}$ is the number of row-constant Young tableaux of shape
%$\lambda$ and content $\mu$.  (See, e.g., \cite{RemTrans}.)
%Like monomial class functions and traces, these functions are not characters.
 
%It is known that 
For any $\sn$-class function $\theta$ belonging to the bases above, 
and for any element $z$ of the natural or Kazhdan-Lusztig basis of $\zsn$,
we have $\theta(z) \in \mathbb Z$.
This follows from the linearity of $\theta$ and the fact that 
%irreducible characters can be expressed as
%traces of integer matrices
%Murnaghan-Nakayama algorithm which computes
$\chi^\lambda(w)$ can be expressed as the trace of an integer matrix.
(See, e.g., \cite[Sec.\,2.3]{Sag}.)
%as the sum of a sequence whose components are all $\pm 1$,
%the fact that
%combined with the facts that 
%that the above class function bases belong
%to $\spn_{\mathbb Z}\{ \chi^\lambda \,|\, \lambda \vdash n \}$
%(See \cite{RemTrans}.)
%and that the Kazhdan-Lusztig basis belongs to $\zsn$.
%(See \cite[Sec.\,2.3]{Sag}, \cite{RemTrans}, \cite{KLRepCH}.)
On the other hand, we do not in general have an elementary formula
for the integer $\theta(z)$.
This incomplete understanding of the $\sn$-class functions is unfortunate,
%This is unfortunate, 
since the functions encode much important information about $\sn$.
%our understanding of these functions is incomplete.
%For instance, 
%Perhaps the most basic knowledge
%In particular, 
%we have no general elementary formulas for the evaluation
%of 
%our ability to evaluate these functions at elements of $\zsn$ is limited.
% incomplete.
%Even 
%for an irreducible character $\chi^\lambda$ and a permutation $w \in \sn$,
%For instance, we have no general elementary formula for the evaluations
%$\{ \chi^\lambda(w) \,|\, \lambda \vdash n, w \in \sn \}$.
%the number $\chi^\lambda(w)$.
%although the Murnaghan-Nakayama algorithm 
%We know by
% play an important role 
%in our understanding of $\sn$, 
%our ability to evaluate them at elements of popular bases of $\zsn$ 
%is limited.  
%that 
%Indeed for the most important of these,
%the irreducible $\sn$-characters
%it is known that irreducible $\sn$-characters 
%$\{ \chi^\lambda \,|\, \lambda \vdash n \}$
%satisfy $\chi^\lambda(w) \in \mathbb Z$ for 
%all 
%$w \in \sn$, but 
%Thus for any integer linear combination of these
%\begin{equation*}
%\theta = \sum_{\lambda \vdash n} a_\lambda \chi^\lambda 
%%\in \spn_{\mathbb Z}\{ \chi^\lambda \,|\, \lambda \vdash n \}
%\end{equation*}
%of these
%the irreducible characters, 
%and any element $z \in \zsn$, we have
%$\theta(z) \in \mathbb Z$ as well.
For some class functions and basis elements,
we may associate sets $R$, $S$ to the pair $(\theta, z)$
to 
%provide a combinatorial interpretation of 
combinatorially interpret
the integer 
$\theta(z)$ as
%\begin{equation*}
%$\theta(z) = 
$(-1)^{|S|}|R|$,
or simply as $|R|$ if $\theta(z) \in \mathbb N$.
%\end{equation*}
We summarize 
%known 
results and open problems in the following table.

\hhhsp
\begin{center}
\newcolumntype{R}{>{$}c<{$}}
%\begin{tabularx}{166.5mm}{|R|R|R|R|R|}%
\begin{tabularx}{155.8mm}{|R|R|R|R|R|}%
\hline
%& & & &\\
& & & &\\
& & & &\\
\theta
%: \zsn \rightarrow \mathbb C 
& \theta(w) \in \mathbb N \mathrm{?} 
%& \begin{matrix} \mbox{interpretation of} \\ 
%\mbox{$\theta(w)^{\phantom{\hat T}}\nTksp$ as $(-1)^{|S|}|R|$?} \end{matrix}
& \begin{matrix} \mbox{interpretation} \\ 
\mbox{of $\theta(w)^{\phantom{\hat T}}\nTksp$}\\ 
\mbox{as $(-1)^{|S|}|R|$?} \end{matrix}
& \theta(C'_w(1)) \in \mathbb N \mathrm{?}  
& \begin{matrix} \mbox{interpretation of} \\ 
%\mbox{$\theta(C'_w(1))$ as $|R|^{\phantom{\hat T}}\nTksp$ for} \\ 
%\mbox{$w$ avoiding $312^{\phantom{\hat T}}\nTksp$?}\end{matrix} \\
\mbox{$\theta(C'_w(1))$ as $|R|^{\phantom{\hat T}}\nTksp$} \\ 
\mbox{for $w^{\phantom{\hat T}}\nTksp$ avoiding} \\
\mbox{$3412^{\phantom{\hat T}}\nTksp$ and $4231$?}\end{matrix} \\
& & & &\\
& & & &\\
\hline 
%& & & &\\
& & & &\\
& & & &\\
%& & & &\\
\eta^\lambda & \mbox{yes} & \mbox{yes} & \mbox{yes} & \mbox{yes} \\
& & & &\\
& & & &\\
\epsilon^\lambda & \mbox{no} & \mbox{yes} & \mbox{yes} & \mbox{yes} \\
& & & &\\
& & & &\\
\chi^\lambda & \mbox{no} & \mbox{open} & \mbox{yes} & \mbox{yes}\\
& & & &\\
& & & &\\
\psi^\lambda & \mbox{yes}  & \mbox{yes} & \mbox{yes} & \mbox{yes} \\
& & & &\\
& & & &\\
\phi^\lambda & \mbox{no} & \mbox{yes} & \mbox{conj.\ by Stembridge, Haiman} & \mbox{open} \\
& & & &\\
& & & &\\
\hline
\end{tabularx}
\end{center}
%\vspace{2mm}
\ssp

%\hsp
%\begin{center}
%\newcolumntype{R}{>{$}c<{$}}
%\begin{tabularx}{133mm}{|R|R|R|R|R|}%
%\hline
%& & & &\\
%\theta
%& \theta(w) \in \mathbb N \mathrm{?} 
%& \begin{matrix} \mbox{interpretation of} \\ \mbox{$\theta(w)$ as $(-1)^{|S|}|R%|$?} \end{matrix}
%& \theta(C'_w(1)) \in \mathbb N \mathrm{?}  
%& \begin{matrix} \mbox{interpretation of} \\ \mbox{$\theta(C'_w(1))$ as $|R|$ f%or} \\ \mbox{$w$ avoiding $312$?}\end{matrix} \\
%& & & &\\
%\hline 
%& & & &\\
%\eta^\lambda & \mbox{yes} & \mbox{yes} & \mbox{yes} & \mbox{yes} \\
%& & & &\\
%\epsilon^\lambda & \mbox{no} & \mbox{yes} & \mbox{yes} & \mbox{yes} \\
%& & & &\\
%\chi^\lambda & \mbox{no} & \mbox{open} & \mbox{yes} & \mbox{yes}\\
%& & & &\\
%\psi^\lambda & \mbox{yes}  & \mbox{yes} & \mbox{yes} & \mbox{yes} \\
%& & & &\\
% \phi^\lambda & \mbox{no} & \mbox{yes} & \begin{matrix} \mbox{conjectured by} \%\ \mbox{Stembridge, Haiman} \end{matrix} & \mbox{open} \\
%& & & &\\
%\hline
%\end{tabularx}
%\end{center}
%%\vspace{2mm}
%\ssp

For the above 
%known 
combinatorial interpretations of $\theta(w)$, 
see 
%\cite[Sec.\,2]{RemTrans}.
\cite{RemTrans}.
The number $\chi^\lambda(w)$ may be computed by the well-known algorithm
of Murnaghan and Nakayama (See, e.g., \cite[Ch.\,7]{StanEC2}.)  
%Otherwise, $\chi^\lambda(w)$ 
but has no conjectured expression 
of the type stated above. 
Interpretations of $\theta(C'_w(1))$ are not known for general $w \in \sn$,
but nonnegativity follows from work of 
Haiman~\cite{HaimanHecke} and Stembridge~\cite{StemImm}.
Interpretations of $\eta^\lambda(C'_w(1))$, 
$\epsilon^\lambda(C'_w(1))$, 
$\chi^\lambda(C'_w(1))$,
$\psi^\lambda(C'_w(1))$,
for $w$ avoiding the 
patterns $3412$, $4231$
%pattern $312$
follow via straightforward arguments from 
results of the fourth author 
{\cite[Thms.\ 4.3, 5.4]{SkanNNDCB}} and 
%the results of various authors,
%the work of other researchers,
others,
notably 
Gasharov~\cite{GashInc},
Karlin-MacGregor~\cite{KMG}, 
Lindstr\"om~\cite{LinVRep}, 
Littlewood~\cite{LittlewoodTGC}, 
Merris-Watkins~\cite{MerWatIneq},
Stanley-Stembridge~\cite{StanStemIJT}, \cite{StemImm}.
These will be discussed in Section~\ref{s:pathposet}.
%The conjecture $\phi^\lambda(C'_w(1)) \in \mathbb N$ is due to 
%Haiman~\cite{HaimanHecke} and Stembridge~\cite{StemConj}.
There is no conjectured combinatorial interpretation of 
%the numbers
$\phi^\lambda(C'_w(1))$, even for $w$ 
\avoidingp,
%avoiding the 
%pattern $312$
%patterns $3412$, $4231$,
%although 
but
interpretations have been given 
for particular partitions $\lambda$
by Stembridge~\cite{StemConj},
several of the authors~\cite{CSSkanPathTab},
and Wolfgang~\cite{WolfgangThesis}.
The problem of 
%combinatorially 
interpreting $\theta(C'_w(1))$
when $w$ does not \avoidp\ is open.
%($\star$ Cite Remmel paper, Stembridge, Haiman, Stanley.)  
%($\star$ Discuss some specific entries.)
%($\star$ Mention special cases of $\phi^\lambda$ proved by Stembridge,
%Skandera et.\ al.)

%It is known that 
Our understanding of $\hnq$-traces is even less complete. 
%than
%our understanding of 
%that of $\sn$-class functions.
%It is known that 
We know that irreducible $\hnq$-characters 
$\{ \chi_q^\lambda \,|\, \lambda \vdash n \}$
satisfy $\chi_q^\lambda(T_w) \in \mathbb Z[q]$ for all $w \in \sn$, since
%This follows from the fact that
$\chi^\lambda_q(T_w)$ can be expressed as the trace of a $\mathbb Z[q]$ 
matrix~\cite{KLRepCH}.
%matrix with entries in $\mathbb Z[q]$~\cite{KLRepCH}.
%(See, e.g., \cite[Sec.\,2.3]{Sag}.)
%and that a $q$-extension of the Murnaghan-Nakayama algorithm computes each
%such polynomial as a ($\star$ signed sum of powers of $q$?) ($\star$ cite?)
%As before, it follows that 
Thus for any element $\theta_q$ of the mentioned $\hnq$-trace bases 
and any element $z \in \spn_{\mathbb Z[q]} \{ T_w \,|\, w \in \sn \}$,
%$z$ in the $\mathbb Z[q]$-span of the natural basis of $\hnq$,
we have $\theta_q(z) \in \mathbb Z[q]$ as well.
For instance, elements of a modified Kazhdan-Lusztig basis 
%elements do not belong to this span, their
%$\qp12$-multiples 
$\{ \qew C'_w(q) \,|\, w \in \sn \}$ belong to this span.
%which
%the irreducible
%$\hnq$-characters, 
%and any element 
%$z \in \spn_{\mathbb Z[q]} \{ T_w \,|\, w \in \sn \}$, 
%Unfortunately,
On the other hand, 
we do not have a general elementary formula
for the polynomial $\theta_q(z)$.
This is unfortunate, since $\hnq$-characters are important in the study of 
$\hnq$ and quantum groups. 
In some cases, we may associate sequences 
$(S_k)_{k \geq 0}$, $(R_k)_{\geq 0}$ of sets to the pair $(\theta_q, z)$
to combinatorially interpret
%provide a combinatorial interpretation of the polynomial 
$\theta_q(z)$ as
%\begin{equation*}
%\theta_q(z) = 
$\sum_k (-1)^{|S_k|}|R_k| q^k$,
or simply as $\sum_k |R_k| q^k$ if $\theta_q(z) \in \mathbb N[q]$.
%\end{equation*}
We summarize 
%known 
results and open problems in the following table.

\hhhpsp
\begin{center}
\newcolumntype{R}{>{$}c<{$}}
%\begin{tabularx}{163.9mm}{|R|R|R|R|R|}%
\begin{tabularx}{158.22mm}{|R|R|R|R|R|}%
%\begin{tabularx}{\linewidth}{|R|R|R|R|R|}%
\hline
& & & &\\
& & & &\\
\theta_q \ntnsp
& \theta_q(T_w) \in \mathbb N[q] \mathrm{?}\ntksp 
& \begin{matrix}\mbox{interpretation of}\\ 
\mbox{$\theta_q(T_w)^{\phantom{\hat T}}\nTksp$ as}\\ 
\mbox{$\sum_k^{\phantom{\hat T}} (-1)^{|S_k|}|R_k|q^k$?}\end{matrix}\ntksp
& \theta_q(\qew C'_w(q)) \in \mathbb N[q] \mathrm{?}  
& \begin{matrix} \mbox{interpretation of}\\ 
\mbox{$\theta_q(\qew C'_w(q))^{\phantom{\hat T}}\nTksp$ as}\\ 
\mbox{$\sum_k^{\phantom{\hat T}}\ntksp|R_k|q^k$ for}\\ 
%\mbox{$w$ avoiding $312^{\phantom{\hat T}}\nTksp$?}\end{matrix} \\
\mbox{$w^{\phantom{\hat T}}\nTksp$ avoiding}\\ 
\mbox{$3412^{\phantom{\hat T}}\nTksp$ and $4231$?}\end{matrix} \\
& & & &\\
& & & &\\
\hline 
& & & &\\
& & & &\\
\eta_q^\lambda & \mbox{no} & \mbox{open} & \mbox{yes} & \mbox{stated in Section 5}\\
& & & &\\
& & & &\\
%\vspace{1mm} 
\epsilon_q^\lambda & \mbox{no} & \mbox{open} & \mbox{yes} & \mbox{stated in Section 6}\\
& & & &\\
& & & &\\
%\vspace{1mm} 
\chi_q^\lambda \ntksp & \mbox{no} & \mbox{open} & \mbox{yes} & \mbox{stated in Section 8}\\
& & & &\\
& & & &\\
%\vspace{1mm} 
%\end{matrix} \\
%& & & &\\
\psi_q^\lambda \ntksp & \mbox{no} & \mbox{open} & \mbox{conj.\ by Haiman} & \mbox{stated in Section 9}\\
%\psi_q^\lambda \ntksp & \mbox{no} & \mbox{open} & \mbox{proved in Section 9} & \mbox{stated in Section 9}\\
& & & &\\
& & & &\\
%\vspace{1mm} 
%\end{matrix} \\
\phi_q^\lambda \ntksp & \mbox{no} & \mbox{open} & \mbox{conj.\ by Haiman} & \mbox{open} \\
& & & &\\
& & & &\\
\hline
\end{tabularx}
\end{center}
%\vspace{2mm}
%\ssp

%\hsp
%\begin{center}
%\newcolumntype{R}{>{$}c<{$}}
%\begin{tabularx}{147.5mm}{|R|R|R|R|R|}%
%\hline
%& & & &\\
%\theta_q \ntnsp
%& \theta_q(T_w) \in \mathbb N[q] \mathrm{?}\ntksp 
%& \begin{matrix}\mbox{interpretation of} \\ \mbox{$\theta_q(T_w)$ as}\\ \mbox{$%\sum_k(-1)^{|S_k|}|R_k| q^k$?}\end{matrix}\ntksp
%& \theta(\qew C'_w(q)) \in \mathbb N[q] \mathrm{?}  
%& \begin{matrix} \mbox{interpretation of} \\ \mbox{$\theta_q(\qew C'_w(q))$ as}%\\ \mbox{$\sum_k|R_k|q^k$ for} \\ \mbox{$w$ avoiding $312$?}\end{matrix} \\
%& & & &\\
%\hline 
%& & & &\\
%\eta_q^\lambda & \mbox{yes} & \mbox{open} & \mbox{yes} & \begin{matrix} \mbox{o%pen} \\ \mbox{(conjectured in Section 4)}  %\vspace{1mm} 
%\end{matrix} \\
%& & & &\\
%\epsilon_q^\lambda & \mbox{no} & \mbox{open} & \mbox{yes} & \begin{matrix} \mbo%x{previously open} \\ \mbox{(stated in Section 4)}  %\vspace{1mm} 
%\end{matrix} \\
%& & & &\\
%\chi_q^\lambda \ntksp & \mbox{no} & \mbox{open} & \mbox{yes} & \begin{matrix} \%mbox{previously open} \\ \mbox{(stated in Section 4)}  %\vspace{1mm} 
%\end{matrix} \\
%& & & &\\
%\psi_q^\lambda \ntksp & \mbox{yes}  & \mbox{open} & \mbox{yes} & \begin{matrix}% \mbox{open} \\ \mbox{(conjectured in Section 4)} %\vspace{1mm} 
%\end{matrix} \\
%& & & &\\
%\phi_q^\lambda \ntksp & \mbox{no} & \mbox{open} & \begin{matrix} \mbox{conjecture%d by} \\ \mbox{Haiman}\end{matrix} & \mbox{open} \\
%& & & &\\
%\hline
%\end{tabularx}
%\end{center}
%%\vspace{2mm}
%\ssp

The polynomial $\chi_q^\lambda(T_w)$, 
and therefore all polynomials $\theta_q(T_w)$ above,
may be computed via a $q$-extension of the Murnaghan-Nakayama algorithm,
developed 
%(See, e.g., \cite{KerVerCFR}, \cite{KingWRepTr}, \cite{RamFrob}.)
in~\cite{KerVerCFR}, \cite{KingWRepTr}, \cite{RamFrob}, \cite{VanderAlg}.
However,
%Otherwise, $\theta_q^\lambda(T_w)$ 
none of these has a conjectured expression of the type stated above.
Interpretations of $\theta_q(\qew C'_w(q))$ 
are not known for general $w \in \sn$,
but results concerning containment in $\mathbb N[q]$ 
follow principally from work of Haiman~\cite{HaimanHecke}.
% and Stembridge~\cite{StemImm}.
In all cases above, coefficients of $\theta_q(\qew C'_w(q))$ are symmetric
about $\qew$.
%$\qp{\ell(w)}2$.
% \cite[?]{HaimanHecke}.
They are also unimodal 
for $\theta_q \in \{ \eta_q^\lambda, \epsilon_q^\lambda, \chi_q^\lambda \}$
and conjectured to be so for $\theta_q \in \{ \psi_q^\lambda, \phi_q^\lambda \}$
\cite[Lem.\,1.1, Conj.\,2.1]{HaimanHecke}.
For $w$ \avoidingp,
%pattern $312$,
%patterns $3412$, $4231$,
formulas for 
$\eta_q^\lambda(\qew C'_w(q))$, 
$\epsilon_q^\lambda(\qew C'_w(q))$, 
$\chi_q^\lambda(\qew C'_w(q))$, and 
$\psi_q^\lambda(\qew C'_w(q))$ 
follow from work of 
Athanasiadis~\cite{AthanPSE},
% in Section~\ref{s:hnqinterp}.  
%Work of 
Gasharov~\cite{GashInc},
Shareshian-Wachs~\cite{SWachsChromQF}, and 
the authors.
% then implies
%a formula for $\chi_q^\lambda(\qew C'_w(q))$.
%These results 
%%and conjectures 
%will be discussed in 
%Sections~\ref{s:hnqinterph},
%Sections~\ref{s:hnqinterpe},
%Sections~\ref{s:hnqinterps},
%and Sections~\ref{s:hnqinterpp}.
%, \ref{s:chnqinterp}, respectively.
%Conjectures and results for $\psi_q^\lambda(\qew C'_w(q))$,
%due to the authors, Shareshian-Wachs, and Athanasiadis,
%These results and conjectures 
%will be discussed in 
%Section~\ref{s:chnqinterp}.
%, \ref{s:chnqinterp}, respectively.
%Interpretations of $\eta^\lambda(C'_w(1))$, 
%The conjecture $\phi_q^\lambda(\qew C'_w(q)) \in \mathbb N[q]$ is due to 
%Haiman~\cite{HaimanHecke}.
There is no conjectured combinatorial interpretation of 
%the numbers
$\phi_q^\lambda(\qew C'_w(q))$, even for $w$ avoiding the 
%pattern $312$.
patterns $3412$, $4231$,
although for particular partitions $\lambda$
interpretations are given by 
the authors in Section~\ref{s:hnqinterpm}.
The problem of combinatorially interpreting $\theta_q(\qew C'_w(q))$
when $w$ does not \avoidp\ is open.

%$\star$ Mention that symmetry of coefficients in $\theta_q(\qew C'_w(q))$
%follows from results of Haiman.  Unimodality too, for 
%all but $\psi_q^\lambda$ follow from Haiman. Unimodality of 
%coefficients in $\psi_q^\lambda$ is a special case of a conjecture of Haiman.

Another way to understand the evaluations $\theta(w)$ 
%and $\theta_q(T_w)$
is to define 
a generating function $\imm{\theta}(x)$ 
for $\{ \theta(w) \,|\, w \in \sn \}$
in the polynomial ring $\zxn$. 
Similarly, we may define
a generating function $\imm{\theta_q}(x)$ 
for $\{ \theta_q(T_w) \,|\, w \in \sn \}$
in a certain noncommutative ring $\anq$. 
In some cases these generating functions have simple forms.
We summarize known results in the following tables.

\hhhsp
%\vspace{5mm}
\begin{center}
\newcolumntype{R}{>{$}c<{$}}
%\begin{tabularx}{41mm}{|R|R|}%
\begin{tabularx}{70.2mm}{|R|R|}%
\hline
&\\
&\\
&\\
\theta
%: \zsn \rightarrow \mathbb C 
& \mbox{simple expression for $\imm{\theta}(x)$?} \\
%& \begin{matrix} \mbox{simple expression} \\ \mbox{for $\imm{\theta}(x)$?} \end{matrix} \\
%&\\
&\\
&\\
\hline 
&\\
&\\
&\\
\eta^\lambda & \mbox{yes} \\
&\\
&\\
&\\
\epsilon^\lambda & \mbox{yes} \\
&\\
&\\
&\\
\chi^\lambda & \mbox{open} \\
&\\
&\\
&\\
\psi^\lambda & \mbox{yes} \\
%\wpsi^\lambda & \mbox{yes} \\
&\\
&\\
&\\
\phi^\lambda & \mbox{open} \\
&\\
&\\
&\\
\hline
\end{tabularx}
\qquad \qquad
%\begin{tabularx}{55.3mm}{|R|R|}%
\begin{tabularx}{71.6mm}{|R|R|}%
\hline
&\\
&\\
\theta_q
%: \csn \rightarrow \mathbb C 
%& \begin{matrix} \mbox{simple expression} \\ \mbox{for $\imm{\theta_q}(x)$?} \end{matrix} \\
& \mbox{simple expression for $\imm{\theta_q}(x)$?} \\
&\\
&\\
\hline 
&\\
&\\
\eta_q^\lambda & \mbox{yes} \\
&\\
&\\
\epsilon_q^\lambda & \mbox{yes} \\
&\\
&\\
\chi_q^\lambda & \mbox{open} \\
&\\
&\\
%\psi_q^\lambda & \mbox{conj.\ in Section 2} \\
\psi_q^\lambda & \mbox{open} \\
%\wpsi_q^\lambda & \mbox{?} \\
&\\
&\\
\phi_q^\lambda & \mbox{open} \\
&\\
&\\
\hline
\end{tabularx}
\end{center}
\vspace{3mm}
\ssp

Simple expressions for 
$\imm{\eta^\lambda}(x)$
and $\imm{\epsilon^\lambda}(x)$
are due to Littlewood~\cite{LittlewoodTGC} and Merris-Watkins~\cite{MerWatIneq},
and a simple expression for $\imm{\psi^\lambda}(x)$ follows immediately from a
standard definition of $\psi$.  
An expression for $\imm{\chi^\lambda}(x)$ as a coefficient of a 
generating function in two sets of variables was given by 
Goulden-Jackson~\cite{GJMaster}.
There is no conjectured simple formula for 
$\imm{\phi^\lambda}(x)$,
although 
a simple formula 
for particular partitions $\lambda$
was stated by Stembridge~\cite{StemConj}. 
%and several of the authors~\cite{CSSkanPathTab}.
%These results will be discussed in Section~\ref{s:imm}.
Simple expressions for 
$\imm{\eta_q^\lambda}(x)$
and $\imm{\epsilon_q^\lambda}(x)$
are due to the fourth author and Konvalinka~\cite{KSkanQGJ},
as is a (less simple) 
expression for $\imm{\chi_q^\lambda}(x)$ as a coefficient in a 
generating function in two sets of variables.
%A simple expression for $\imm{\psi_q^\lambda}(x)$ is conjectured by the authors.
%in Section~\ref{s:?}.
%These results and conjecture will also be discussed in Section~\ref{s:imm}.

%($\star$ Rewrite this.)
In Section~\ref{s:imm} 
we discuss 
%known descriptions of $\sn$-class functions $\theta$ in terms of
generating functions $\imm{\theta}(x)$ 
%in the ring $\cxn$ for
for $\sn$-class functions $\theta$ 
and 
generating functions $\imm{\theta_q}(x)$ 
for $\hnq$-traces $\theta_q$.
These generating functions belong to the ring
$\zxn$ and to a certain $q$-analog $\anq$ 
%$\cxn$ and to a certain quantum analog $\anq$ 
of $\zxn$ known as the {\em quantum matrix bialgebra}.
%We also 
%We also give a combinatorial interpretation of the entries of the
%transition matrices relating certain bases of $\anq$.
%state one result
%We present some results about that too.
In Section~\ref{s:dsn}
%In Section~\ref{s:sninterp} 
we relate these generating functions to structures 
called {\em zig-zag networks}~\cite{SkanNNDCB},
which serve as combinatorial interpretations for
Kazhdan-Lusztig basis elements indexed by permutations \avoidingp.
In Section~\ref{s:pathposet} we introduce a partial order on paths in
zig-zag networks, and show how this poset and results in the literature lead 
%simply (?) 
to combinatorial interpretations for the evaluations
of $\sn$-class functions at the above Kazhdan-Lusztig basis elements
of $\zsn$.
%In Sections~\ref{s:hnqinterph} -- \ref{s:hnqinterpm}
For the remainder of the article,
we concentrate on combinatorial interpretations of trace evaulations
of the form $\theta_q(\qew C'_w(q))$ where $w$ \avoidsp.
In Sections~\ref{s:hnqinterph} -- \ref{s:hnqinterpe} 
we %give new combinatorial interpretations of the trace evaluations
interpret
$\eta_q^\lambda(\qew C'_w(q))$
and $\epsilon_q^\lambda(\qew C'_w(q))$. 
%when $w$ \avoidsp.
In Section~\ref{s:chromsf} we recall the relationship between
$\sn$-class functions and Stanley's chromatic symmetric functions,
and prove that $\hnq$-traces are similarly related to the
Shareshian-Wachs chromatic quasisymmetric functions.
In Sections~\ref{s:hnqinterps} -- \ref{s:hnqinterpp} 
we use results of Shareshian-Wachs and Athanasiadis to
interpret %state combinatorial interpretations of the trace evaluations
$\chi_q^\lambda(\qew C'_w(q))$
and $\psi_q^\lambda(\qew C'_w(q))$. 
One of our interpretations proves a formula conjectured by 
Haiman~\cite[Conj.\,4.1]{HaimanHecke}.
%for $w$ \avoidingp.  
%We also conjecture three additional interpretations
%of $\psi_q^\lambda(\qew C'_w(q))$ for $w$ \avoidingp, and prove that
%these interpretations are equivalent for $w$ avoiding $312$.
%$\chi_q^\lambda(\qew C'_w(q))$, 
%when $w$ \avoidsp.
%descriptions ($\star$ ?) of the class functions and traces
%in terms of
%generating functions in the rings $\mathbb C \otimes \Lambda$
%and $\cqq \otimes \Lambda$ of symmetric functions having coefficients
%in $\mathbb C$ and $\cqq$.
Finally, in Section~\ref{s:hnqinterpm} we state several results 
%and conjectures
%combinatorial
%interpretations for the trace evaluations
%$\psi_q^\lambda(\qew C'_w(q))$, 
%and special cases of
concerning 
%the evaluations 
$\phi_q^\lambda(\qew C'_w(q))$.
%when $w$ \avoidsp.
%draw connections to posets and to the
%chromatic quasisymmetric functions of Shareshian and Wachs.

%\bp\label{p:charinterpbasis}
%Find combinatorial interpretations of the nonnegative integers
%$\epsilon^\lambda(C'_w(1))$, 
%$\eta^\lambda(C'_w(1))$, 
%$\phi^\lambda(C'_w(1))$, 
%$\chi^\lambda(C'_w(1))$,
%and of the 
%Laurent polynomials 
%$\epsilon_q^\lambda(C'_w(q))$, 
%$\eta_q^\lambda(C'_w(q))$, 
%$\phi_q^\lambda(C'_w(q))$, 
%$\chi_q^\lambda(C'_w(q))$
%in $\smash{\qp12}$ 
%which hold for all $w \in \sn$ and $\lambda \vdash n$.
%\ep

%The $\sn$-class functions and $\hnq$-traces above are 
%related to symmetric functions by the {\em Frobenius characteristic formula}.  
%(See, e.g., \cite{RamFrob}, \cite{Sag}.) 
%This formula allows one to define $\sn$-class functions and $\hnq$-traces
%corresponding to any symmetric functions, such as 
%the {\em $k$-Schur functions}
%and {\em dual $k$-Schur functions} 
%of Lapointe, Lascoux, and Morse~\cite{LLMTableau}. 
%Let $\chi^{\lambda,k}$ and $\phi^{\lambda,k}$ denote the $\sn$-class functions
%corresponding to the $k$-Schur symmetric functions and the dual
%$k$-Schur symmetric functions.
%Let $\chi_q^{\lambda,k}$ and $\phi_q^{\lambda,k}$ denote the corresponding
%$\hnq$-traces.

%\bp\label{p:kscharinterpbasis}
%Find combinatorial interpretations of the 
%integers
%$\chi^{\lambda,k}(C'_w(1))$,
%$\phi^{\lambda,k}(C'_w(1))$, 
%and of the Laurent polynomials
%$\phi_q^{\lambda,k}(C'_w(q))$, 
%$\chi_q^{\lambda,k}(C'_w(q))$
%in $\smash{\qp12}$
%which hold for all $w \in \sn$ and $\lambda \vdash n$.
%\ep

%\bp
%Give a new interpretation of $\imm{\pi^\lambda}(A)$ for power sum immanants.
%\ep


%\bp\label{p:kschurinterp}
%Let $\chi^{\lambda,k}$ and $\phi^{\lambda,k}$ be the class functions
%corresponding by the Frobenius characteristic function to
%the {\em $k$-Schur functions} and {\em dual $k$-Schur functions} 
%of Morse and Lapointe
%($\star$ cite).
%Give a combinatorial interpretation of 
%$\chi^{\lambda,k}(C'_w(1))$ and
%$\phi^{\lambda,k}(C'_w(1))$. 
%\ep


%has a combinatorial
%interpretation 
%these,
%including the other $\sn$-class functions defined in Section ?,
%have the same property.
%%all integer linear combinations of 
%In some casesfor any $\sn$-class function $\theta$ which is
%an integer linear combination of irreducible characters,

%\vspace{-10mm}
%\ssec{Basic definitions and difficult open problems}\label{ss:repbasic}


\section{Generating functions for $\theta(w)$ and $\theta_q(T_w)$ when 
$\theta$ ($\theta_q$) is fixed}\label{s:imm}
% and $w$ varies}\label{s:imm}

%($\star$ Say something general about this point of view of generating functions.)

%\newpage
%\section{Total nonnegativity and the polynomial ring $\cxn$}
%\label{s:tnn}
%, and $\sn$-characters}\label{s:tnn}

%\vspace{-10mm}
%\ssec{The classical case and total nonnegativity}
%\label{ss:tnnbasic}

For a fixed $\sn$-class function $\theta$, we
create a generating function for $\{ \theta(w) \,|\, w \in \sn \}$
by writing a matrix of variables $x = (x_{i,j})_{i,j \in [n]}$
%, $\cx \smash{\defeq} \cxn$, 
and defining 
%we will use the polynomial ring 
%$\cxn$ to 
\begin{equation*}
%\label{eq:immdef}
\imm \theta(x) \defeq \sum_{w \in \sn} \theta(w) \permmon xw \ \in \ 
%\cx \smash{\defeq} \cxn,
\zx \defeq \zxn,
%\quad
%\mathrm{ in } 
%\quad
\end{equation*}
where $w_1 \cdots w_n$ is the one-line notation of $w$. 
%(See, e.g., \cite{StanEC1}.)
We call this polynomial the 
{\em $\theta$-immanant}.  The sign character 
%($\sgn: w \mapsto (-1)^{\ell(w)}$)
($w \mapsto (-1)^{\ell(w)}$)
immanant and trivial character 
%($\triv: w \mapsto 1$)
($w \mapsto 1$)
immanant are the determinant and permanent,
\begin{equation*}
\det(x) = \sum_{w \in \sn} (-1)^{\ell(w)} \permmon xw,
\qquad
\per(x) = \sum_{w \in \sn} \permmon xw.
\end{equation*}
Simple formulas for the $\epsilon^\lambda$-immanants 
and $\eta^\lambda$-immanants 
%$\{ \imm{\epsilon^\lambda}(x) \,|\, \lambda \vdash n \}$
%and
%$\{ \imm{\eta^\lambda}(x) \,|\, \lambda \vdash n \}$
employ
%in terms of 
determinants and permanents of square submatrices $x_{I,J}$ of $x$,
%.  We write
%$x_{I,J}$ is the $I,J$ submatrix of $x$,
\begin{equation*}
%x_{I,J} \underset{\text{def}}= (x_{i,j})_{i \in I, j \in J},
x_{I,J} \defeq (x_{i,j})_{i \in I, j \in J},
\qquad 
%\begin{gathered}
I,J \subset [n] \defeq \{1,\dotsc, n\},
\qquad 
|I| = |J|.
%,\\
%\qquad |I| = |J|.
%\end{gathered}
\end{equation*}
%of $x$.  %(See \cite{MerWatIneq}.)
%(where these results are attributed to Littlewood?)
% or other paper by those authors.)
In particular, 
%for $\lambda = (\lambda_1, \dotsc, \lambda_r) \vdash n$ we have
we have the 
Littlewood-Merris-Watkins identities~\cite{LittlewoodTGC}, \cite{MerWatIneq} 
%showed that for
%and 
%In particular, Littlewood~\cite{LittlewoodTGC} and 
%Merris and Watkins~\cite{MerWatIneq} showed that for
%$\lambda = (\lambda_1, \dotsc, \lambda_r) \vdash n$, we have
\begin{equation}\label{eq:inducedimmssubmat}
%\begin{gathered}
\imm{\epsilon^\lambda}(x) = 
\nTtksp
\sum_{(I_1, \dotsc, I_r)} 
\nTksp
\det(x_{I_1, I_1}) \cdots \det(x_{I_r, I_r}), \quad
%\\
\imm{\eta^\lambda}(x) = 
\nTtksp
\sum_{(I_1, \dotsc, I_r)} 
\nTksp
\per(x_{I_1, I_1}) \cdots \per(x_{I_r, I_r}),
%\end{gathered}
\end{equation}
where $\lambda = (\lambda_1, \dotsc, \lambda_r) \vdash n$ and
%where 
the sums are over 
all sequences 
%$(I_1, \dotsc, I_r)$
of pairwise disjoint subsets of $[n]$ satisfying $|I_j| = \lambda_j$.
%\end{thm}
We will call such a sequence an {\em ordered set partition} of $[n]$ 
{\em of type $\lambda$,} and will sometimes write $I \vdash [n]$
and $\ell(\lambda) = r$.

A simple
%somewhat less attractive, but still rather simple, 
formula for the $\psi^\lambda$-immanant 
involves
%relies upon 
a sum over all permutations of cycle type $\lambda$.  
%(See, e.g., \cite{Sag}, \cite{StanEC1} for definitions.)
Specifically, we have
\begin{equation}\label{eq:pimm}
\imm{\psi^\lambda}(x) = z_\lambda 
\nTksp \sumsb{w\\ \ctype{w} = \lambda} \nTksp
\permmon xw,
\end{equation}
where $z_\lambda$ is the product
%\begin{equation*}
%z_\lambda = 
$1^{\alpha_1} 2^{\alpha_2} \cdots n^{\alpha_n} \alpha_1! \cdots \alpha_n!$,
%\end{equation*}
and $\lambda$ has $\alpha_i$ components (or {\em parts}) 
equal to $i$ for $i = 1, \dotsc, n$.  
%(See, e.g., \cite{Sag}, \cite{StanEC1}.)  
No such simple formulas are known for the 
$\chi^\lambda$-immanants or $\phi^\lambda$-immanants in general,
although Stembridge gave a formula~\cite[Thm.\,2.8]{StemConj}
for $\imm{\phi^\lambda}(x)$ when $\lambda$ is the rectangular partition
$(k)^r = (k,\dotsc,k)$.  In this case we have
%$\lambda_1 = \cdots = \lambda_r = k$,
\begin{equation}\label{eq:stemmonrect}
\imm{\phi^{(k)^r}}(x) = \sum_{(I_1,\dotsc, I_k)} 
\det(x_{I_1,I_2}) \det(x_{I_2,I_3}) \cdots
%\det(x_{I_{k-1},I_k}) 
\det(x_{I_k,I_1}),
\end{equation}
where the sum is over all ordered set partitions 
%of sequences 
%$(I_1, \dotsc, I_k)$
%of pairwise disjoint subsets of 
of $[n] = [kr]$ 
%satisfying $|I_j| = r$.
of type $(r)^k$.
(See \cite{GJMaster} for an expression for $\imm{\chi^\lambda}(x)$ 
as a coefficient of a generating function in two sets of variables.)
%$\lambda^\tr = r^k$.

%($\star$ Define $[n]$ somewhere earlier.)

%($\star$ Mention Stembridge formula for rectangular monomial immanants?)

For a fixed $\hnq$-trace $\theta_q$, we
create a generating function for $\{ \theta_q(T_w) \,|\, w \in \sn \}$
as before,
%by writing $x = (x_{i,j})$ as before, 
except that we interpret
%but interpreting
polynomials in 
$x = (x_{i,j})$ 
%$x = (x_{i,j})_{i,j \in [n]}$ 
%as before, 
%$x$ 
as elements of
%a certain noncommutative $\cqq$-algebra
%noncommutative analog
% we define 
%$\anq$ which we call 
the {\em quantum matrix bialgebra} $\anq$,
%For each $n \geq 0$, let the {\em quantum polynomial ring} 
%$\anq$ is 
the noncommutative $\zqq$-algebra
%$\mathbb{C}[\qp12,\qm12]$-algebra
generated by $n^2$ variables 
%$x = (x_{1,1}, \dotsc, x_{n,n})$,
$x_{1,1}, \dotsc, x_{n,n}$,
subject to the relations 
%on every $2 \times 2$ submatrix
%\begin{equation}
%x_{\{i,j\},\{k,\ell\}} = 
%\begin{bmatrix}
%x_{i,k} & x_{i,\ell} \\
%x_{j,k} & x_{j,\ell}
%\end{bmatrix}
%\end{equation}
%\begin{equation}\label{eq:qringrelations}
%\begin{aligned}
%x_{i,\ell}x_{i,k} &= \qp12x_{i,k} x_{i,\ell}, \\
%x_{j,k}x_{i,k} &= \qp12x_{i,k} x_{j,k}, \\
%x_{j,k} x_{i,\ell} &= x_{i,\ell}x_{j,k},\\
%x_{j,\ell} x_{i,k} &= x_{i,k} x_{j,\ell} + (\qdiff) x_{i,\ell}x_{j,k},
%\end{aligned}
%\end{equation}
\begin{equation}\label{eq:qringrelations}
\begin{alignedat}{2}
x_{i,\ell}x_{i,k} &= \smash{\qp12}x_{i,k} x_{i,\ell}, &\qquad
x_{j,k} x_{i,\ell} &= x_{i,\ell}x_{j,k}\\
x_{j,k}x_{i,k} &= \smash{\qp12}x_{i,k} x_{j,k} &\qquad
x_{j,\ell} x_{i,k} &= x_{i,k} x_{j,\ell} + \smash{(\qdiff)} x_{i,\ell}x_{j,k},
\end{alignedat}
\end{equation}
for all indices $1 \leq i < j \leq n$ and $1 \leq k < \ell \leq n$.
As a $\smash{\zqq}$-module, $\anq$ has a natural basis of monomials 
$x_{\ell_1,m_1} \cdots x_{\ell_r,m_r}$ in which index pairs
appear in lexicographic order.  
%We will call this basis the {\em natural basis} of $\anq$.
%We can use
The relations (\ref{eq:qringrelations}) 
allow one
to express other monomials in terms of this natural basis.
%As a $\mathbb{C}[\qp12,\qm12]$-module, 
%$\anq$ is spanned by monomials in lexicographic order, 
%of the polynomial ring $\cxn$ 
%conjunction with the
%quantum coordinate ring 
%$\oqslnc$ 
%of the special linear group. 
%$SL(n,\mathbb{C})$.  
%A noncommutative analog of the polynomial ring $\cx$, 
%We call $\anq$ the {\em quantum polynomial ring},
%In particular, we have the definition
%isomorphism
%\begin{equation}\label{eq:oqslncdef}
%\oqslnc \defeq \anq/(\qdet(x) - 1),
%\end{equation}
%where $\qdet(x)$ isintro

As a generating function for $\{ \theta_q(T_w) \,|\, w \in \sn \}$,
we define
\begin{equation*}
\imm{\theta_q}(x) \defeq \sum_{w \in \sn} \theta_q(T_w) \qiew \permmon xw
\end{equation*}
in $\anq$, and call this 
%polynomial 
the 
{\em $\theta_q$-immanant}.  The $\hnq$ sign character 
%($\sgn_q: T_w \mapsto (-1)^{\ell(w)}$)
($T_w \mapsto (-1)^{\ell(w)}$)
immanant and 
%$\hnq$ 
trivial character 
%($\triv_q: T_w \mapsto q^{\ell(w)}$)
($T_w \mapsto q^{\ell(w)}$)
immanant are called the {\em quantum determinant} and {\em quantum permanent},
\begin{equation*}
%\begin{gathered}
%\imm{\sgn_q}(x) = 
\qdet(x) = \sum_{w \in \sn} (-\qm12)^{\ell(w)} \permmon xw,
\qquad
%\\
%\imm{\triv_q}(x) = 
\qper(x) = \sum_{w \in \sn} (\qp12)^{\ell(w)} \permmon xw.
%\end{gathered}
\end{equation*}
%Note that the specialization of 
%$\oqslnc$ and 
Specializing $\anq$ at $\smash{\qp12}=1$ gives
the commutative polynomial ring $\zx$, 
%where the images of %elements
with elements $\qdet(x)$ and $\qper(x)$ 
%at $\smash{\qp12}=1$, we obtain 
%coordinate ring of $SL(n,\mathbb{C})$ and the 
%are 
specializing to
the classical determinant $\det(x)$ and permanent $\per(x)$.
Simple formulas for the $\epsilon_q^\lambda$-immanants 
and $\eta_q^\lambda$-immanants 
%$\{ \imm{\epsilon^\lambda}(x) \,|\, \lambda \vdash n \}$
%and
%$\{ \imm{\eta^\lambda}(x) \,|\, \lambda \vdash n \}$
employ
quantum 
%in terms of 
determinants and 
quantum 
permanents of submatrices
%$x_{I,J}$ is the $I,J$ submatrix of $x$,
%\begin{equation*}
%x_{I,J} 
%\underset{\text{def}}= (x_{i,j})_{i \in I, j \in J}
%\end{equation*}
of $x$.  %(See \cite{MerWatIneq}.)
%(where these results are attributed to Littlewood?)
% or other paper by those authors.)
In particular, the fourth author and Konvalinka~\cite[Thm.~5.4]{KSkanQGJ}
%Littlewood~\cite{LittlewoodTGC} and 
%Merris and Watkins showed~\cite[Sec.\,1, 6]{MerWatIneq} 
%Merris and Watkins~\cite{MerWatIneq} showed that for
%\begin{thm}\label{t:lmw}
%Fix a partition 
%identies proved by Konvalinka and the candidate
%\begin{thm}\label{t:KSkan}
proved quantum analogs of the Littlewood-Merris-Watkins identities 
%where the sums are as in 
%in 
(\ref{eq:inducedimmssubmat}),
%that for $\lambda = (\lambda_1, \dotsc, \lambda_r) \vdash n$ we have
\begin{equation}\label{eq:qinducedimmssubmat}
%\begin{gathered}
\ntksp
\imm{\epsilon_q^\lambda}(x) 
= 
\nTtksp
\sum_{(I_1,\dotsc,I_r)} 
\nTtksp
\qdet(x_{I_1,I_1}) \cdots \qdet(x_{I_r,I_r}),
%\\ 
\quad
\imm{\eta_q^\lambda}(x) 
= 
\nTtksp
\sum_{(I_1,\dotsc,I_r)} 
\nTtksp
\qper(x_{I_1,I_1}) \cdots \qper(x_{I_r,I_r}),
\ntksp 
%\end{gathered}
\end{equation}
where the sums are as in (\ref{eq:inducedimmssubmat}).
See \cite{KSkanQGJ} for an expression for the
$\chi_q^\lambda$-immanant as a coefficient of a generating function 
in two sets of variables.
No formulas are known for the 
$\psi_q^\lambda$- 
or $\phi_q^\lambda$-
%, or $\psi_q^\lambda$-
immanants.
It would be interesting to state a $q$-analog of (\ref{eq:stemmonrect}) 
for rectangular partitions $\lambda = (k)^r$.

%($\star$ Conjecture analog of 
%Stembridge formula for rectangular monomial immanants?
%Use this conjecture to conjecture a formula for quantum power sum
%immanants?)

%To obtain values of $\epsilon_q^\lambda(T_w)$ and $\eta_q^\lambda(T_w)$
%from (\ref{eq:qinducedimmssubmat}), one must use the relations
%(\ref{eq:qringrelations}) to expand in the natural basis of 
%(the zero-weight space 
%$\spn\{ \permmon xw \,|\, w \in \sn \}$ of)
%$\anq$.
%For this purpose, it is helpful to combinatorially interpret
%the coefficients arising as entries in the transition matrix relating
%the bases $\mathcal B_u = \{ \doublepermmon xuv \,|\, v \in \sn \}$ 
%and the natural basis $\{ \permmon xw \,|\, w \in \sn \}$.
%These were obtained by Lambright and the fourth author in~\cite{LSkanCFDPR}.
%($\star$ Is this necessary?)
%To combinatorially interpret the evaluations 
%$\{ \epsilon_q^\lambda(\qew C'_w(q)) \,|\, \lambda \vdash n\}$ when
%$w$ avoids the pattern $312$, 
%%patterns $3412$, $4231$, 
%we prove a stronger result.
%\begin{thm}
%Fix $u, w \in \sn$ with $u \leq w$,
%%and $u$ minimal in a Young subgroup of $\sn$,
%and let $s_{i_1} \cdots s_{i_\ell}$ be the right-to-left lexicographically 
%greatest reduced expression for $u$. 
%Choose an index $k \leq \ell + 1$ and define
%$u' = s_{i_{k-1}} \cdots s_{i_1}u$, 
%$w' = s_{i_{k-1}} \cdots s_{i_1}w$.
%Then 
%we have
%\begin{equation*}
%\doublepermmon xuw = \sum_{v \in \sn} t_{u, w', v}(\qdiff)\doublepermmon x{u'}v,
%\end{equation*}
%where 
%$\{t_{u,w',v}(q) \,|\, v \in \sn \}$ are polynomials in $\mathbb N[q]$.
%Moreover, the coefficient of $q^b$ in $t_{u,w',v}(q)$ is equal to
%the number of sequences $(\pi^{(0)}, \dotsc, \pi^{(k-1)})$ of permutations
%satisfying
%\begin{enumerate}
%\item $\smash{\pi^{0} = w}$, $\smash{\pi^{(k-1)} = v}$,
%\item $\smash{\pi^{(j)} \in \{ s_{i_j}\pi^{(j-1)}, \pi^{(j-1)} \}}$ 
%for $j = 1,\dotsc, k-1$,
%\item $\smash{\pi^{(j)} = s_{i_j}\pi^{(j-1)}}$ if $\smash{s_{i_j} \pi^{(j-1)} > \pi^{(j-1)}}$,
%\item $\smash{\pi^{(j)} = \pi^{(j-1)}}$ for exactly $b$ values of $j$.
%\end{enumerate}
%\end{thm}
%\begin{proof}
%Omitted.
%\end{proof}
%\noindent
%We may think of each sequence $(\pi^{(0)},\dotsc,\pi^{(k-1)})$
%in the above proof as a $(k-1)$-step walk from $w$ to $v$ 
%in the weak order on $\sn$.  After visiting $\pi^{(j)} \in \sn$,
%we may either revisit this permutation or move to $s_{i_j}\pi^{(j)}$,
%with the latter option being mandatory if $s_{i_j}$ 
%is a left ascent for $\pi^{(j)}$.
%($\star$ Explain walks in Bruhat order and include picture?)

%($\star$ Mention $\sigma$ maps.  Here or elsewhere?)

%($\star$ Definitely include an example.)

\section{Planar 
%Zig-zag 
networks
%, descending star networks, 
and path matrices}\label{s:dsn}
%combinatorial 
%interpretations
%of 
%$\sn$-
%class functions}\label{s:sninterp}

Call a directed planar graph $G$ a
%an English (French)
{\em planar network of order $n$} if it
is acyclic and 
may be embedded in a disc with $2n$ boundary vertices labeled
clockwise as 
{\em source $1, \dotsc,$ source $n$} 
(with indegrees of $0$) and 
{\em sink $n, \dotsc,$ sink $1$}
(with outdegrees of $0$).
In figures, we will draw 
sources on the left and sinks on the right,
implicitly labeled $1, \dotsc, n$ 
%and $n$ sinks on the right labeled $1, \dotsc, n$,
%with both sides labeled from top-to-bottom 
from bottom to top.
Edges will be implicitly oriented from left to right.
Given a planar network $G$, 
define the {\em path matrix} $B = B(G) = (b_{i,j})$ of $G$ by
%Define the {\em path matrix} $B = B(G) = (b_{i,j})$ of $G$ by
\begin{equation}\label{eq:pathmatrix}
b_{i,j} = \text{ number of paths in $G$ from source $i$ to sink $j$}.
\end{equation}
%It is known that the
The 
path matrix of any planar network is totally nonnegative,
%(TNN), 
i.e., 
%that 
each square submatrix has nonnegative determinant.
%every minor of this matrix is nonnegative.
Specifically, for sets
$I = \{ i_1, \dotsc, i_k \}$, $J = \{ j_1, \dotsc, j_k \} \subset [n]$,
and the corresponding submatrix
%with elements 
%of both sets 
%listed in increasing order,
%define 
$B_{I,J} \defeq (b_{i,j})_{i \in I, j \in J}$, 
% with $|I| = |J| = k$,
we have that $\det(B_{I,J})$ is equal to the number of families 
$(\pi_1, \dotsc, \pi_k)$ of mutually nonintersecting paths from
sources $i_1, \dotsc, i_k$ (respectively) 
to sinks $j_1, \dotsc, j_k$ (respectively).
This fact is known as {\em Lindstr\"om's Lemma}~\cite{LinVRep}.
(See also \cite{KMG}.)
%The converse of Lindstr\"om's Lemma is also true:
%every totally nonnegative matrix is the path
%matrix of some (not unique) planar network.  (Technically, 
%this converse requires that we assign
%nonnegative {\em weights} to the edges of a planar network and
%that we modify the definition of
%path matrix appropriately.  We ignore these details in this document.)
%($\star$ State $\sn$ presentation and conventions.)
We will call two planar networks $G_1$, $G_2$ {\em isomorphic} 
and will write $G_1 \cong G_2$ 
if $B(G_1) = B(G_2)$.

For example, consider two isomorphic planar networks,
their common path matrix $B$ and its submatrix $B_{23,13}$:
\begin{equation*}
\raisebox{-8mm}{
\includegraphics[height=18mm]{xfigures/pnet1}}\ ,
\qquad
\raisebox{-8mm}{
\includegraphics[height=18mm]{xfigures/pnet2}}\ ,
\qquad
B = \begin{bmatrix}
1 & 1 & 0\\
1 & 2 & 1\\
0 & 1 & 2
\end{bmatrix},
\qquad
B_{23,13} = \begin{bmatrix}
1 & 1\\
0 & 2
\end{bmatrix}.
\end{equation*}
We can interpret $\det(B_{23,13}) = 2$ as counting the two path families 
\begin{equation*}
\raisebox{-8mm}{
\includegraphics[height=18mm]{xfigures/pnet3}}\ ,
\qquad
\raisebox{-8mm}{
\includegraphics[height=18mm]{xfigures/pnet4}}\ ,
\end{equation*}
from sources $\{2, 3\}$ to sinks $\{1, 3\}$ in the first network.

An easy fact about planar networks is the following.
\begin{obs}\label{o:pnetintersect}
Let $G$ be a planar network of order $n$ 
and assume that for some indices $i, i', j, j' \in [n]$,
$G$ contains a path $\pi_i$ from source $i$ to sink $j$ and 
a path $\pi_{i'}$ from source $i'$ to sink $j'$.  
If these two paths intersect,
then $G$ also contains a path from source $i$ to sink $j'$ and
a path from source $i'$ to sink $j$.  
If $i' < i$ and $j < j'$ then the paths $\pi_i$ and $\pi_{i'}$ cross.
\end{obs}


%Call a planar network of order $n$ a {\em bijective skeleton} if it is a union
%of $n$ source-to-sink paths.  
%It is easy to see that a bijective skeleton can be covered by at least
%one path family of type $e$, and by exactly one path family of type $e$
%in which paths do not cross.
%($\star$ Cut?) Clearly a bijective 
%path family can cover an entire
%planar network $G$ only if $G$ is a bijective skeleton.
%It is also easy to see that there is a bijective skeleton is cover
%by a path family of type $e$.

For $[a,b]$ a subinterval of $[n]$,
% = \{1, \dotsc, n\}$, 
let $G_{[a,b]}$
be the planar network
%{\em star network}
consisting of $a-1$ horizontal edges
from sources $1, \dotsc, a-1$ to corresponding sinks,
a ``star'' of $b-a+1$ 
edges from sources $a, \dotsc, b$ to an intermediate vertex
and $b-a+1$ more edges from this vertex to sinks $a,\dotsc,b$,
and $n-b$ more horizontal edges
from sources $b+1,\dotsc,n$ to corresponding sinks. 
For $n = 4$, there are seven such networks:
$G_{[1,4]}$, 
$G_{[2,4]}$, 
$G_{[1,3]}$, 
$G_{[3,4]}$, 
$G_{[2,3]}$, 
$G_{[1,2]}$, 
%and 
$G_{[1,1]} = \cdots = G_{[4,4]}$, respectively,
\begin{equation*}
\raisebox{-6mm}{
\includegraphics[height=12mm]{xfigures/stars}}.
%\includegraphics[width=50mm]{xfigures/stars.eps}}.
\end{equation*}
%In French notation, 
%These are 
%$G_{[1,4]}$, 
%$G_{[2,4]}$, 
%$G_{[1,3]}$, 
%$G_{[3,4]}$, 
%$G_{[2,3]}$, 
%$G_{[1,2]}$, 
%%and 
%$G_{[1,1]} = \cdots = G_{[4,4]}$, respectively.

Given planar networks $G$, $H$ of order $n$, in which all sources have
outdegree $1$ and all sinks have indegree $1$, define
$G \circ H$ to be the concatenation of $G$ and $H$, 
where for $i = 1,\dotsc, n$, sink $i$ of $G$ is dropped,
source $i$ of $H$ is dropped, and 
the unique edge in $G$ from vertex $x$ to sink $i$ 
and the unique edge in $H$ from source $i$ to vertex $y$ 
are merged to form a single edge from $x$ to $y$ in $G \circ H$.
%Call the concatenation of finitely many star networks
%a {\em generalized wiring diagram}.  
%We will say that two generalized wiring diagrams are 
%{\em isomorphic} if they are isomorphic as directed graphs.  
Note that for star networks $G_{[c_1,d_1]}$, $G_{[c_2,d_2]}$ 
indexed by nonintersecting intervals, the two concatenations
$G_{[c_1,d_1]} \circ G_{[c_2,d_2]}$ and 
$G_{[c_2,d_2]} \circ G_{[c_1,d_1]}$ are isomorphic.
% as directed graphs.
 
We will be interested in concatenations
%generalized wiring diagrams
\begin{equation}\label{eq:concat}
G = G_{[c_1,d_1]} \circ \cdots \circ G_{[c_t,d_t]}
\end{equation}
%satisfying the conditions
such that
\begin{enumerate}
%in which 
\item the sequence $([c_1,d_1], \dotsc, [c_t,d_t])$
consists of $t$ distinct, pairwise nonnesting intervals, 
%with the additional
%property 
%of pairwise nonnesting intervals satisfy
%the additional condition 
%that 
\item for $i < j < k$, 
% the following conditions.
%\begin{enumerate}
%\item No pair of intervals is nested.  
%(The intervals are pairwise nonnested.)
%(For $i \neq j$ we have $(c_j - c_i)(d_j - d_i) > 0$.)
%(For $i \neq j$ we have $c_i < c_j$ and $d_i < d_j$ or
%$c_i > c_j$ and $d_i > d_j$.)
%%
%\item For $i < j < k$, 
if $[c_i,d_i] \cap [c_j,d_j] \neq \emptyset$
and $[c_j,d_j] \cap [c_k,d_k] \neq \emptyset$, then we have
$c_i < c_j < c_k$ 
(and $d_i < d_j < d_k$) 
or $c_i > c_j > c_k$
(and $d_i > d_j > d_k$).
\end{enumerate}
%Given the concatenation (\ref{eq:concat}), 
We define a relation $\precdot$
on the set of intervals 
appearing 
in the concatenation (\ref{eq:concat})
by declaring $[c_i,d_i] \precdot [c_k,d_k]$ 
if 
\begin{enumerate}
\item $i < k$, 
\item $[c_i,d_i] \cap [c_k,d_k] \neq \emptyset$, 
%and 
\item $[c_i,d_i] \cap [c_j,d_j] \cap [c_k,d_k] = \emptyset$ 
for $j = i+1, \dotsc, k-1$.
\end{enumerate}
%The relation $\precdot$ 
This is the covering relation of a partial order $\preceq$.
%transitive closure $\prec$ of $\precdot$ defines a partial order 
% on the set of intervals.  
%(See, e.g., \cite[Ch.\,3]{StanEC1} for definitions.)
%Let $\prec$ be 
%, and 
%and by taking
%the transitive closure of $\precdot$.  
%by declaring $[c_i,d_i] \preceq [c_j,d_j]$ if $i \leq j$ and 
%$[c_i,d_i] \cap [c_j,d_j] \neq \emptyset$, and by taking
%the transitive closure of this relation.  
%Let $\precdot$ denote the
%If the Hasse diagram of this poset (as an undirected graph)
%is a disjoint union of paths,
%union of maximal paths, any two of which have an empty intersection,
%or else share only their maximal element or only their minimal element,
%then call it a {\em zig-zag poset}.
%($\star$ End of wrong material.)
%($\star$ The following is probably now irrelevant.)
%For $n=4$, the zig-zag posets are 
%\begin{equation}\label{eq:zigzag}
%\raisebox{-6mm}{
%\includegraphics[width=109mm]{zzp.eps}}\ .
%\end{equation}
%($\star$ End of probably irrelevant material.)
%It is easy to see that

For each planar network $G$ 
of the form
%$G = G_{[c_1,d_1]} \circ \cdots \circ G_{[c_t,d_t]}$
%such that $\preceq$ defines a zig-zag poset on the set of intervals, 
%labeled as in 
(\ref{eq:concat})
%and 
satisfying the conditions following (\ref{eq:concat}), we 
%use the 
%covering 
%relation $\precdot$ 
%of the partial order $\preceq$ 
%to 
define a related planar network $F$ by 
considering each covering 
%modifying $G$ as follows.
%For each 
%covering 
pair $[c_i,d_i] \precdot [c_j,d_j]$
% with $[c_i, d_i]$ covered by $[c_j,d_j]$
%in the poset and 
with $| [c_i,d_i] \cap [c_j,d_j] | = k$, 
and deleting all but one of the $k$ paths 
from the central vertex of $G_{[c_i,d_i]}$ to the central vertex of $G_{[c_j,d_j]}$.
% to a single path between the vertices.  
Following \cite{SkanNNDCB}, we call the resulting network $F$ 
a {\em zig-zag network}.
%($\star$ This procedure is not precise: there can be several zig-zag networks
%for a concatenation. They are all isomorphic.  Fix this somehow?
%Mention it and claim that it doesn't need to be fixed?
%Several concatenations can lead to the same zig-zag network isomorphism 
%class too.  Define a canonical concatenation?  Define isomorphism of zig-zag
%networks to be isomorphism when the only recognized vertices are sources,
%sinks, and central star vertices?)
In the special case that we have
%satisfying
%\begin{equation*}
$c_1 > \cdots > c_t$,
% and $d_1 > \cdots > d_t$, 
we call $F$ a {\em descending star network}.
%and the concatenation
%$G_{[c_1,d_1]} \circ \cdots \circ G_{[c_t,d_t]}$ of corresponding star networks.
The descending star networks (up to isomorphism) of order $4$ are
\begin{equation}\label{eq:xfigures2}
\raisebox{-6mm}{
\includegraphics[height=12mm]{xfigures/xfigures2}}\ .
%\includegraphics[width=135mm]{xfigures/xfigures.eps}}.
\end{equation}
The zig-zag networks 
%(up to isomorphism) 
of order $4$ 
which are not descending star networks are
\begin{equation}\label{eq:xfigureszz}
\raisebox{-6mm}{
\includegraphics[height=12mm]{xfigures/xfigureszz}}\ .
\end{equation}
%\includegraphics[width=135mm]{xfigures/xfigures.eps}}.

%For each such planar network $G$, we define a related planar network $F$
%by modifying $G$ as follows. 
%For $i = 1, \dotsc, t-1$, if the intersection $[c_{i+1},d_{i+1}] \cap [c_i,d_i]$
%has cardinality $k > 1$, then collapse the $k$ paths from 
%the central vertex of $G_{[c_{i+1},d_{i+1}]}$ to 
%the central vertex of $G_{[c_i,d_i]}$, creating 
%a single path between these vertices.
%Call $F$ a {\em descending star network}.
%For $n=4$, the decending star networks are
%\begin{equation}\label{eq:xfigures2}
%\raisebox{-6mm}{
%\includegraphics[height=12mm]{xfigures2}}.
%\end{equation}
It is easy to see that if $F$ is a zig-zag network of order $n$,
then there is at most one interval in the concatenation 
(\ref{eq:concat}) containing $n$, and this interval must be
maximal or minimal (or both) in the partial order $\preceq$.

It was shown in \cite[Thm.\ 3.5, Lem.\ 5.3]{SkanNNDCB} 
that zig-zag networks of order $n$ correspond bijectively to 
\pavoiding permutations in $\sn$.  To summarize this bijection,
% as follows.
we let $F$ be the zig-zag 
%descending star 
network obtained from 
%which corresponds as before (\ref{eq:xfigures}) to 
the concatenation $G$ in (\ref{eq:concat}), and construct
another concatenation of star networks as follows.
%$G = G_{[c_1,d_1]} \circ \cdots \circ G_{[c_t,d_t]}$.
For $i = 1, \dotsc, t-1$, if the interval $[c_i,d_i]$ is covered by $[c_j,d_j]$
in the order $\preceq$ and if $|[c_i,d_i] \cap [c_j,d_j]| > 1$, then insert 
$G_{[c_i,d_i] \cap [c_j,d_j]}$ 
immediately after $G_{[c_i,d_i]}$ 
in the current concatenation.  
(If $[c_i,d_i]$ is also covered by a second interval $[c_k,d_k]$, 
then $G_{[c_i,d_i] \cap [c_k,d_k]}$ may be inserted before 
or after $G_{[c_i,d_i] \cap [c_j,d_j]}$.)
Call the resulting augmented network $G'$.
%Modify $G$ to create the related network 
%\begin{equation*}
%G' \smash{\defeq} G_{[c_1,d_1]} \circ G_{[c_1,d_1] \cap [c_2,d_2]} \circ G_{[c_%2,d_2]} 
%\circ \cdots \circ
%G_{[c_{t-1},d_{t-1}]} \circ G_{[c_{t-1},d_{t-1}] \cap [c_t,d_t]} \circ G_{[c_t,%d_t]} 
%\end{equation*}
%by inserting $G_{[c_i,d_i] \cap [c_{i+1},d_{i+1}]}$ between $G_{[c_i,d_i]}$
%and $G_{[c_{i+1},d_{i+1}]}$ for $i=1,\dotsc,t-1$.  
Now visually follow paths from sources to sinks in $G'$, 
passing ``straight'' through each star,
%intersection,
to create a \pavoiding permutation $w \in \sn$. 
See \cite[Sec.\,3]{SkanNNDCB} for a description of the inverse
of this bijection, which maps $w$ to $G'$.
%complete a bijection to \pavoiding permutations in $\sn$.
%$312$-avoiding permutations in $\sn$.
%\end{proof}
We will let $F_w$ and $G'_w$ denote the zig-zag network 
and augmented star network corresponding to a fixed
\pavoiding permutation $w$, and we will let $w(F)$ denote the
\pavoiding permutation corresponding to a fixed zig-zag network $F$.
%If $F$ and $w$ correspond by this bijection, we will sometimes denote
%the 

For example, 
%when $n = 4$ and $F$ corresponds to $G = G_{[2,4]} \circ G_{[1,3]}$,
%we construct $G' = G_{[2,4]} \circ G_{[2,3]} \circ G_{[1,3]}$
%and obtain the $312$-avoiding permutation $w = w(F) = 3421$:
%when $n = 9$ and $F$ corresponds to 
suppose that 
%when 
$F$ is the zig-zag network obtained from the concatenation
%$n = 9$ and $F$ corresponds to 
$G = G_{[3,7]} \circ G_{[5,8]} \circ G_{[8,9]} \circ G_{[1,2]} \circ G_{[2,4]}$
of star networks of order $9$.
Drawing the poset $\preceq$ on these intervals from left to right, we have
%these intervals form a zig-zag poset 
%which is the union of three chains  
%$[3,7] \prec [5,8] \prec [8,9]$, 
%$[3,7] \prec [2,4]$, 
%$[1,2] \prec [2,4]$,
%whose Hasse diagram forms a path as an undirected graph,
\begin{equation*}
\raisebox{-12mm}{
\includegraphics[height=40mm]{xfigures/izz}}\ .
\end{equation*}
Since the only covering pairs which intersect at more than an endpoint are
$[3,7] \precdot [5,8]$ and $[3,7] \precdot [2,4]$, we construct
$G'$ by inserting
$G_{[3,7] \cap [5,8]} = G_{[5,7]}$ and $G_{[3,7] \cap [2,4]} = G_{[3,4]}$ after $G_{[3,7]}$.
%We thus construct 
Thus we have
$G' = G_{[3,7]} \circ G_{[5,7]} \circ G_{[3,4]} \circ G_{[5,8]} \circ G_{[8,9]} \circ G_{[1,2]} \circ G_{[2,4]}$,
and we obtain the \pavoiding permutation $w = w(F) = 419763258$:
%we insert $G_{[2,3]}$ between $G_{[2,4]}$ and $G_{[1,3]}$ to obtain
%\hsp
\begin{equation*}
%\label{eq:xfiguresextrax}
%\smash{
%\begin{center}
%\qquad \qquad \qquad
%F = \raisebox{-6mm}{
%\includegraphics[height=12mm]{xstars1}}, \qquad
%\includegraphics[height=12mm]{xfigures/xstars1}}, \qquad
F = \ntnsp \raisebox{-17mm}{
\includegraphics[height=34mm]{xfigures/bigzzd}}\ , \quad
%G = \raisebox{-6mm}{
%\includegraphics[height=12mm]{xstars2}}, \qquad
%\includegraphics[height=12mm]{xfigures/xstars2}}, \qquad
G = \raisebox{-17mm}{
\includegraphics[height=34mm]{xfigures/bigzz}}\ , \quad
G' = \raisebox{-17mm}{
%\includegraphics[height=12.25mm]{xstars3}}, \qquad
\includegraphics[height=34mm]{xfigures/bigzzp}}\ , \quad
%\includegraphics[height=12.25mm]{xfigures/xstars3}}, \qquad
%w = \binom{1234}{3421}.
w = \binom{123456789}{419763258}.
%\end{center}
%}
%w = 3421.
%\thinspace \thinspace 
%\quad \qquad \qquad 
%\raisebox{-10mm}{$\square$}
\end{equation*}
%\end{proof}
%\ssp

Note that we have $G_{[3,4]} \circ G_{[5,7]} \cong G_{[5,7]} \circ G_{[3,4]}$,
since the intervals $[3,4]$, $[5,7]$ do not overlap.
In general, we have the following.
\begin{obs}\label{o:firstorlast}
If $[c_{i_1}, d_{i_1}], \dotsc, [c_{i_t},d_{i_t}]$ is a linear extension
of the poset $\preceq$ defined in terms of the concatenation 
%$G$ in
(\ref{eq:concat}), then we have
\begin{equation*}
G_{[c_1,d_1]} \circ \cdots \circ G_{[c_t,d_t]} 
\cong
G_{[c_{i_1},d_{i_1}]} \circ \cdots \circ G_{[c_{i_t},d_{i_t}]},
\end{equation*}
and the corresponding zig-zag networks are isomorphic as well.
\end{obs} 

Call a sequence $\pi = (\pi_1, \dotsc, \pi_n)$ of source-to-sink paths
in a planar network $G$ of order $n$ a {\em path family}.
%{\em bijective path family}
% of type $w$} 
We will always assume that path $\pi_i$ begins at source $i$.
If for some $w \in \sn$ with one-line notation $w_1 \cdots w_n$, 
each component path $\pi_i$ 
%begins at source $i$ 
%and 
terminates at sink $w_i$, 
we will say that $\pi$ has {\em type $w$} 
and we will write $\mathrm{type}(\pi) = w$.  
If the union of the paths of $\pi$ is equal to $G$,
we will say that $\pi$ {\em covers} $G$.
For example, the planar network $G_{[2,4]} \circ G_{[1,3]}$ 
can be covered by many different path families, 
including two of type $e$, and two of type $2341$:
\begin{equation}\label{eq:cover}
\raisebox{-10mm}{
\includegraphics[height=20mm]{xfigures/cover}}.
\end{equation}

The result \cite[Lem.\,5.3]{SkanNNDCB} states that 
%each zig-zag network
%is a bijective skeleton, 
%(defined earlier?), 
%and that 
the path families covering a zig-zag network $F_w$ 
correspond bijectively to elements of
a principal order ideal in the Bruhat order:
%and describes path families covering $F_w$ as follows.
\begin{thm}\label{t:lem53nndcb}
Fix $v, w \in \sn$ with $w$ \avoidingp.
There is a unique path family of type $v$ covering $F_w$ if and only if
we have $v \leq w$.  Otherwise, there is no such path family.
\end{thm}
Thus $w$ is the unique Bruhat-maximal permutation for which some path
family of type  $w$ covers $F_w$.
It follows also that there is exactly one path in $F_w$ from any source $i$ 
to the corresponding sink $i$, and 
%.  There is 
at most one path 
%in $F_w$ 
from source $i$ to sink $j \neq i$. 
%there is exactly one path.  From any source to any
%{\em other} sink there is at most one path, i.e., $b_{i,j} \in \{0,1\}$.)
Whether or not such a path exists 
%in a zig-zag network from a particular source to a particular sink 
may be determined by the intervals in
the corresponding concatenation of star networks 
and the partial order $\preceq$.

\begin{obs}\label{o:sourcetosink}
Let $F$ be a zig-zag network which corresponds to the concatenation
(\ref{eq:concat}).  There exists a path in $F$ from source $i$ to sink $j$
if and only if we have one of the following.
\begin{enumerate}
\item $i, j \in [c_k,d_k]$ for some $k$.
\item $i \in [c_k, d_k]$, $j \in [c_\ell, d_\ell]$ and 
$[c_k,d_k] \prec [c_\ell, d_\ell]$ for some $k, \ell$.
\end{enumerate}
\end{obs}
It is easy to see that if $F$ is a descending star 
network with sources $i$ and $j$ belonging to the same connected component, 
then the second condition in Observation~\ref{o:sourcetosink} 
is equivalent to the inequality $i > j$.  
%($\star$ This was originally stated without the connectivity assumption.
%Is the old incorrect ``fact'' used somewhere?)
This fact allows us to refine Observation~\ref{o:pnetintersect} slightly.
\begin{lem}\label{l:dsnintersect}
%\begin{lem}\label{l:cyclintersect}
Let $\pi_{\ell_1}$, $\pi_{\ell_2}$ be paths in a descending star network $F$
from sources $\ell_1 < \ell_2$ to sinks $m_1, m_2$, respectively.
%$\pi = (\pi_1, \dotsc, \pi_n)$ be a path family covering a descending
%star network $F$, and fix indices $\ell_1 < \ell_2$.
Then the paths $\pi_{\ell_1}$ and $\pi_{\ell_2}$ intersect if and only if
there exists a path in $F$ from source $\ell_1$ to sink $m_2$.
\end{lem}
\begin{proof}
Assume that $F$ has order $n$ and 
corresponds to the concatenation (\ref{eq:concat}) of star networks.  
Let $x_1, \dotsc, x_t$ be the vertices in $F$
corresponding to the central vertices of the $t$ star networks in 
(\ref{eq:concat}), and 
%$G_{[c_1,d_1]}, \dotsc,$ 
for any index $j \in [n]$, 
let $f(j)$ and $g(j)$ denote the indices of the 
first and last intervals, respectively,
in $[c_1, d_1], \dotsc, [c_t, d_t]$ to contain $j$.
Then for any indices $\ell, m \in [n]$, 
the unique path in $F$ from source $\ell$ to sink $m$
contains the vertices $x_{f(\ell)}, \dotsc, x_{g(m)}$.

It is easy to see that if the intersection 
of $\pi_{\ell_1}$ and $\pi_{\ell_2}$ is nonempty, then there is a path
in $F$ from source $\ell_1$ to sink $m_2$.
%By Observation~\ref{o:sourcetosink},
Now suppose that there is a path in $F$ 
from source $\ell_1$ to sink $m_2$.
%if and only if
By Observation~\ref{o:sourcetosink},
we have $m_2 < \ell_1$ or we have an interval $[c_k, d_k]$ containing
both $\ell_1$ and $m_2$.
% belong to a common interval
%$[c_k, d_k]$ for some $k$.
Either case implies that we have $f(\ell_2) \leq f(\ell_1) \leq g(m_2)$,
and the paths $\pi_{\ell_1}$ and $\pi_{\ell_2}$ share the vertex $x_{f(\ell_1)}$.
\end{proof}
%Assume that $F$ has order $n$ and 
%corresponds to the concatenation (\ref{eq:concat}) of star networks.  
%Let $x_1, \dotsc, x_t$ be the vertices in $F$
%corresponding to the central vertices of the $t$ star networks in 
%(\ref{eq:concat}), and 
%%$G_{[c_1,d_1]}, \dotsc,$ 
%for any index $j \in [n]$, 
%let $f(\ell)$ and $g(\ell)$ denote the indices of the 
%first and last intervals, respectively,
%in $[c_1, d_1], \dotsc, [c_t, d_t]$ to contain $\ell$.
%Then the unique path in $F$ from source $\ell$ to sink $m$
%contains the vertices $x_{f(\ell)}, \dotsc, x_{g(m)}$.
%%By Observation~\ref{o:sourcetosink},
%Suppose that there is a path in $F$ 
%from source $\ell_1$ to sink $m_2$.
%%if and only if
%By Observation~\ref{o:sourcetosink},
%we have $m_2 < \ell_1$ or we have an interval $[c_k, d_k]$ containing
%both $\ell_1$ and $m_2$.
%% belong to a common interval
%%$[c_k, d_k]$ for some $k$.
%Either case implies that we have $f(\ell_2) \leq f(\ell_1) \leq g(m_2)$,
%and the paths $\pi_{\ell_1}$ and $\pi_{\ell_2}$ share the vertex
%$x_{f(\ell_1)}$.
%Now suppose that there is no path in $F$ from source $\ell_1$ to sink $m_2$.
%By Observation~\ref{o:sourcetosink},
%we have that $m_2 > \ell_1$ and that no interval $[c_k, d_k]$ contains
%both $\ell_1$ and $m_2$.  This implies that 
%$g(m_2) < f(\ell_1)$.
%Thus $\pi_{\ell_2}$ passes through vertices
%$x_{f(\ell_2)}, \dotsc, x_{g(m_2)}$, 
%terminating with the subpath
%from vertex $x_{g(m_2)}$ to sink $m_2$,
%%is the concatenation paths from 
%%source $\ell_2$ to vertex $x_{f(\ell_2)}$, 
%%and 
%while
%$\pi_{\ell_1}$ begins with the subpath from 
%%is the concatenation paths from 
%source $\ell_1$ to vertex $x_{f(\ell_1)}$, 
%passing through vertices $x_{f(\ell_1)}, \dotsc, x_{g(m_1)}$, 
%and finally sink $m_1$.
%%and vertex $x_{g(m_1)}$ to sink $m_1$,
%%Since $g(m_2) < f(\ell_1)$, 
%It follows that the two paths do not intersect.
%\end{proof}
%Assume that for some indices $i, i', j, j'$, 
%a zig-zag network $F$ 
%contains a path $\pi_i$ from source $i$ to sink $j$
%and a path $\pi_{i'}$ from source $i'$ to sink $j'$.
%Then the following statements are equivalent.
%\begin{enumerate}
%\item $\pi_i$ and $\pi_{i'}$ intersect.
%\item There is a path from source $i$ to sink $j'$ in $F$.
%\item There is a path from source $i'$ to sink $j$ in $F$.
%\end{enumerate}
%\end{obs}
%let $U$ be an 
%$F_w$-tableau 
%%of shape $(n)$ 
%containing the path family
%$(\pi_1, \dotsc, \pi_n)$, and fix indices $\ell_1 < \ell_2$.  
%Then the paths $\pi_{\ell_1}$ and $\pi_{\ell_2}$ intersect if and only if 
%there exists a path in $F_w$ from $\ell_1$ to the sink of $\pi_{\ell_2}$.
%\end{lem}
%\begin{proof}
%Omitted.
%\end{proof}

%($\star$ Mention that in a zig-zag network $F_w$, for any two paths in
%the unique path family of type $w$, if the paths intersect, they must cross?)

%($\star$ Make this a proof of a proposition?)


%\noindent
%For each $312$-avoiding permutation $w \in \sn$,
%let $F_w$ denote the descending star network corresponding to $w$ by
%the bijection in the proof of Proposition~\ref{p:dsncatalan}.
%($\star$ Omit?) Every descending star network $F_w$ is a bijective skeleton, 
%and for every $v \leq w$ in the Bruhat order, there is exactly one bijective
%path family $\pi$ of type $v$ which covers $F_w$.
%($\star$ 
%Mention that from any source to the corresponding sink, 
%there is exactly one path.  From any source to any
%{\em other} sink there is at most one path, i.e., $b_{i,j} \in \{0,1\}$.)

The subset of zig-zag networks which are descending star networks
can be characterized using pattern avoidance.
\begin{thm}\label{t:dsn312}
Let $v \in \sn$ \avoidp.  Then $v$ avoids the pattern $312$ if and only if
$F_v$ is a descending star network.
\end{thm}
\begin{proof}
Let $G$ be the concatenation (\ref{eq:concat}) of star networks which
leads to $F_v$, and define $G'$ as in the example above.

Suppose first that $F_v$ is not a descending star network.
%and let $\pi = (\pi_1,\dotsc, \pi_n)$ be the unique path family
%of type $v$ covering $F_v$.
Then in the concatenation (\ref{eq:concat}) 
there exists an index
$i$ such that the interval $[c_i,d_i]$ is minimal in the poset $\preceq$,
and an index $j > i$ such that 
$c_i < c_j$ and $[c_i,d_i] \precdot [c_j,d_j]$. 
Considering the relationship between $G'$ and $v$, one sees 
that we have $v_{d_i} < c_j$ and that for some index
$\ell \geq d_j$ we have $v_\ell = c_j$.
%By Proposition~\ref{p:nocrossnointersect}, the paths $\pi_{c_j}$ and $\pi_\ell$
%do not intersect.
On the other hand, since $G'$ is constructed by inserting $G_{[c_j,d_i]}$ 
between $G_{[c_i,d_i]}$ and $G_{[c_j,d_j]}$,
we have that $v_{c_i} \geq d_j$.
Thus the indices $c_i < d_i < \ell$ satisfy 
%$v_{c_i} > v_{\ell} > v_{d_i}$
$v_{d_i} < v_{\ell} < v_{c_i}$
and $v$ does not avoid the pattern $312$.

%Now suppose that $v$ contains a subword which matches the pattern $312$, i.e.,
%some indices $i < j < k$ satisfy $v_j < v_k < v_i$.
%Let $\pi = (\pi_1,\dotsc, \pi_n)$ be the unique path family of type $v$
%which covers $F_v$.  Since the sources and sinks of paths $\pi_j$, $\pi_k$
%satisfy $j < k$ and $v_j < v_k$, the paths do not intersect.
%On the other hand, $\pi_i$ necessarily intersects $\pi_j$ before it intersects
%$\pi_k$

Now suppose that $F_v$ is a descending star network.
We claim that if $F_v$ arises from the concatenation (\ref{eq:concat}),
then $v$ avoids the pattern $312$ and satisfies $v_{c_1} > \cdots > v_{d_1}$.
%It is easy to see that this is true for descending star networks 
By inspection of (\ref{eq:xfigures2}), 
this is true for descending star networks of orders $1$ -- $4$.
Now assume this to be true for
each descending star network $F_w$ of order $1, \dotsc, n-1$, and 
%the permutation $w$ avoids the pattern $312$.
let $F_v$ be a descending star network of order $n$. 
%arising from the concatenation (\ref{eq:concat}).
If in the first interval of (\ref{eq:concat})
we have $d_1 < n$, then $v_n = n$ and $F_{v_1 \cdots v_{n-1}}$ 
is a descending star network of order $n-1$.
Thus $F_v$ has the claimed properties.
%It follows that $v$ avoids the pattern $312$.  
If in the first interval of (\ref{eq:concat}) we have $d_1 = n$, then
consider the descending star network $F_w$ arising from the concatenation
$G_{[c_2,d_2]} \circ \cdots \circ G_{[c_t,d_t]}$.
Then
%we have
$v = s_{[c_1,d_1]} s_{[c_1,d_2]} w$, where $s_{[a,b]}$ is the unique Bruhat-maximal
permutation for which a path family of this type 
covers the star network $G_{[a,b]}$.
By 
%induction,
the above argument, 
$w$ avoids the pattern $312$, and 
satisfies $w_{c_2} > \cdots > w_{d_2}$, 
and $w_i = i$ for $i = d_2+1,\dotsc, n$.
It follows that the letters in positions $c_1, \dotsc, n$ of $s_{[c_1,d_2]} w$
form an increasing sequence, and that $v$ satisfies 
%$v_i = w_i$ for $i = 1, \dotsc, c_1 - 1$, and
$v_{c_1} > \cdots > v_n$.
Thus the subword $v_1 \cdots v_{c_1-1} = w_1 \cdots w_{c_1-1}$ 
avoids the pattern $312$, as does the subword $v_{c_1} \cdots v_n$.
% avoids
%the pattern $12$.
If $v$ contains any subword $z_3z_1z_2$ that matches the pattern $312$,
then $z_3z_1$ must be a subword of 
$v_1 \cdots v_{c_1-1}$ while $z_2$ is a letter of the decreasing word
$v_{c_1} \cdots v_n$.
But in this case, $z_3z_1$ is a subword of $w_1 \cdots w_{c_1-1}$ while
$z_2$ is a letter of $w_{c_1} \cdots w_n$, which contradicts our assumption
that $w$ avoids the pattern $312$.
\end{proof}
%\begin{prop}\label{p:dsncatalan}

It follows that there are 
$\tfrac 1{n+1}\tbinom{2n}n$ descending star networks of order $n$,
since there are this many $312$-avoiding permutations in $\sn$,
all of which \avoidp.
%\end{prop}
%\begin{proof}
%{\bf Proof: }%Omitted.
%($\star$ Move this elsewhere?)
%($\star$ Justify the above with a proposition like the following.)
%\begin{prop}
%Let 
%$w \in \sn$ avoid the pattern $312$.
%Then the path matrix $B$ of $F_w$ 
%satisfies
%\begin{equation}
%\permmon bv = \begin{cases}
%1 &\text{if $v \leq w$},\\
%0 &\text{otherwise}.
%\end{cases}
%\end{equation}
%\end{prop}
%\begin{proof}
%Omitted.
%\end{proof}
%Proofs of the validity of the tableaux interpretations in 
%Proposition~\ref{p:sncfinterp} depend upon 
%a relationship
%between immanants and path matrices.
%To state a $q$-analog of this relationship,
%formula,
Also related to pattern avoidance
are the sizes of the stars in the concatenation (\ref{eq:concat}).

%$G_w$ are closely related to subsequences of $w_1 \cdots w_n$.
\begin{thm}\label{t:kstar}
%Let $F_w$ be a zig-zag network.
Let $w \in \sn$ \avoidp.
%, and let $(\pi_1,\dotsc,\pi_n)$ be the
%unique path family of type $e$ covering $F_w$.
Then $w_1 \cdots w_n$ contains a decreasing subsequence of size $k$
if and only if some interval in the concatenation (\ref{eq:concat})
%for some interval $[i,i+k-1]$, the paths
%$\pi_i, \dotsc, \pi_{i+k-1}$ share a common vertex.
corresponding to $F_w$ has cardinality at least $k$.
\end{thm}
\begin{proof}
Let $w_{i_1}, \dotsc, w_{i_k}$ be a decreasing subsequence of $w$.
Then there is a path in $F_w$ from source $i_j$ to sink $w_{i_j}$
for $j = 1,\dotsc,k$.  These paths pairwise intersect, since
$i_r < i_s$ if and only if $w_{i_r} > w_{i_s}$.
Since $F_w$ is acyclic, these paths must all intersect at a single vertex.
Such a vertex must correspond to the central vertex of 
a star network indexed by an interval of cardinality at least $k$
in the concatenation (\ref{eq:concat}).

The converse is clearly true if we have $t = 1$ in (\ref{eq:concat}).
Suppose that the converse holds for each zig-zag network corresponding
to a concatenation of $t-1$ star networks.
Let $F_w$ correspond to a concatenation (\ref{eq:concat}) of
$t$ star networks, and let $F_v$ correspond to the concatenation
$G_{[c_1, d_1]} \circ \cdots \circ G_{[c_{t-1}, d_{t-1}]}$.
Suppose that some interval $[c_i, d_i]$, $1 \leq i \leq t$, 
%, \dotsc, [c_t, d_t]$
has cardinality at least $k$.
If $i \leq t-1$, then $v$ contains a decreasing subsequence
of size $k$.  
%It is clear that there is a path family of type $v$ which covers $F_w$.  
%Thus $v \leq w$ in the Bruhat order.
By \cite[Cor.\,3.7]{SkanNNDCB}, there is a reduced expression for $w$ 
which consists of a reduced expression for $v$, followed by some reduced
expression $s_{i_1} \cdots s_{i_k}$ for the permutation $v^{-1}w$.
It is well known that each permutation in the sequence 
$(v, vs_{i_1}, vs_{i_1}s_{i_2}, \dotsc, w)$ preserves all inversions of the previous
permutation and introduces one more.  It follows that $w$ also has a
decreasing subsequence of length $k$.
If $i = t$, then apply the above argument to $w^{-1}$, which corresponds
to the concatenation $G_{[c_t, d_t]} \circ \cdots \circ G_{[c_1, d_1]}$.
It is well known that $w$ has a decreasing subsequence of length $k$
if and only if $w^{-1}$ does.
%($\star$ Finish other direction with Sam's proof.)
\end{proof}

We may use path matrices of zig-zag networks to evaluate
$\sn$-class functions at Kazhdan-Lusztig basis elements 
$\{ C'_w(1) \,|\, w \text{ \avoidsp} \}$.
% of $\zsn$
%indexed by \pavoiding permutations.  
Specifically, if $B$ is the path matrix of $F_w$, then 
by \cite[Sec.\,4, Thm.\,5.4]{SkanNNDCB} we have
%we may apply
%a certain duality of the immanant space of $\cx$ and $\sn$
%to Equation (\ref{eq:immdef}) to obtain
\begin{equation}\label{eq:thetaCimmB}
\theta(C'_w(1)) = \imm \theta (B).
\end{equation}
This fact is a crucial ingredient in the proofs of 
Theorems~\ref{t:thetavw}, \ref{t:sncfinterp},
which interpret the evaluations of certain $\sn$-class functions
at Kazhdan-Lusztig basis elements of $\zsn$.

%(See, e.g., \cite[Sec.\,4, Thm.\,5.4]{SkanNNDCB}.) 
In order to interpret the evaluations of $\hnq$-traces 
at Kazhdan-Lusztig basis elements of $\hnq$,
we will prove a $q$-extension of Equation (\ref{eq:thetaCimmB}) 
in Proposition~\ref{p:qimmchar}.
Namely, we will show that path matrices can also be used to  
evaluate $\hnq$-traces
at the (modified) Kazhdan-Lusztig basis elements 
$\{ \qew C'_w(q) \,|\, w \text{ \avoidsp} \}$.
A bit of care is required though: 
the evaluation $\imm{\theta_q}(B)$ does not make sense because
the substitution $x_{i,j} \mapsto b_{i,j}$
does not respect the relations (\ref{eq:qringrelations})
and therefore does 
not give a well-defined map from $\anq$ to $\zqq$.
%Instead of using , 
Thus we define 
%($\star$ first defined here, or in paper with Brittany and Justin, 
%or in paper with Charlie and Justin?) 
a $\zqq$-linear map for each $n \times n$ integer matrix $B$ by
\begin{equation*}
\begin{aligned}
\sigma_B: \anq &\rightarrow \zqq\\
\permmon xv &\mapsto \qev \permmon bv.
\end{aligned}
\end{equation*}
\begin{prop}\label{p:qimmchar}
Let $\theta_q$ be an $\hnq$-trace
and let $w \in \sn$ \avoidp.
%avoid the patterns $312$.
%Let $F_w$ be the planar network corresponding to the $312$-avoiding permutation
%$w \in \sn$, let $B$ be 
Then the path matrix $B$ of $F_w$ 
%as defined in
%(\ref{eq:pathmatrix}),
%and let $\theta_q$ be an $\hnq$-trace.
%Then we have 
satisfies
\begin{equation*}
%\label{eq:qimmcharformula}
\theta_q(\qew C'_w(q)) = \sigma_B(\imm{\theta_q}(x)).
%$
\end{equation*}
\end{prop}
\begin{proof}
The right-hand side is equal to
\begin{equation*}
%\begin{aligned}
%\sigma_A(\imm{\theta_q}(x)) 
%&= 
\sigma_B \Big( \sum_{v \in \sn} \theta_q(T_v) \qiev \permmon xv \Big) 
= \sum_{v \in \sn} \theta_q(T_v) \permmon bv. 
%\end{aligned}
\end{equation*}
By Theorem~\ref{t:lem53nndcb} (\cite[Lem.~5.3]{SkanNNDCB}), 
the product $\permmon bv$ is $1$ 
when $v \leq w$ and is $0$ otherwise.  Thus the above 
expression is equal to
\begin{equation*}
%\sum_{v \leq w} \theta_q(\wT_v) \qev = 
\sum_{v \leq w}\theta_q(T_v) 
= \theta_q \Big( \sum_{v \leq w} T_v \Big).
%= \theta_q(\qew C'_w(q)).
\end{equation*}
Since $w$ \avoidsp, \cite{LakSan} implies that the Kazhdan-Lusztig polynomials 
$\{ P_{v,w}(q) \,|\, v \leq w \}$ are identically 1.
(See 
also 
\cite[Ch.~6]{BilleyLak}.)
% when $v \leq w$ and 
%identically $0$ otherwise.
%the famous result of Lakshmibai and Sandhya
Comparing to (\ref{eq:KLbasis}), we see that the parenthesized sum is equal to 
$\qew C'_w(q)$. 
\end{proof}
Stated from another point of view, Proposition~\ref{p:qimmchar} asserts
that for $w$ \avoidingp, the zig-zag network $F_w$ combinatorially encodes
the modified Kazhdan-Lusztig basis element $\qew C'_w(q)$ in the sense that
\begin{equation*}
\qew C'_w(q) = \sum_{\pi} T_{\mathrm{type}(\pi)},
\end{equation*}
where the sum is over all path families which cover $F_w$.
% and 
%$\mathrm{type}(\pi)$ is the type of $\pi$.

\section{Path posets, planar network tableaux and 
interpretation of $\sn$-class function evaluations}
\label{s:pathposet}
%\section{Interpretations of $\hnq$-traces}

\ssec{Path posets}
In a planar network $G$ of order $n$, the source-to-sink paths
have a natural partial order $Q = Q(G)$.  Given paths $\pi_i$, $\rho_j$,
originating at sources $i$, $j$, respectively,
%and $\rho_j$ is a path originating at source $j$,
% to sink $v_j$,
%then 
we define $\pi_i <_Q \rho_j$ if 
%\begin{enumerate}
%\item $i < j$,
%\item $\pi_i$ and $\rho_j$ never intersect.
%\end{enumerate}
$i < j$ and $\pi_i$ and $\rho_j$ do not intersect.
%Observe that these conditions imply the index of the sink of $\pi_i$ to
%be less than the index of the sink of $\rho_j$.
Let $P(G)$ be the subposet of $Q(G)$ induced by paths
%Observe also that the paths 
whose source and sink indices are equal.
%form a subposet of $Q$.  Call this subposet $P = P(G)$.
%If $F$ is a bijective skeleton of order $n$,
For each zig-zag
%descending star 
network $F_w$, the poset $P(F_w)$ has exactly
$n$ elements:  
%Observe that $P(G)$ will not in general have exactly $n$ elements,
%even if $G$ is a bijective skeleton:
%for some $i$, there may be more than one path from source $i$ to sink $i$.
there is exactly one path from source $i$ to sink $i$, for $i = 1,\dotsc,n$.

The posets $P(G)$ corresponding to the descending star networks $G$ in 
(\ref{eq:xfigures2}) are
\begin{equation}\label{eq:uio}
%\raisebox{-8mm}{
\raisebox{-3mm}{
\includegraphics[width=130mm]{xfigures/uio}}\ .
\end{equation}
These are precisely the 
%four-element 
%posets which contain no induced subposet
%isomorphic to 
$(\mathbf{3}+\mathbf{1})$-free,
%or
$(\mathbf{2}+\mathbf{2})$-free posets on four elements,
where
{\em $(\mathbf{a}+\mathbf{b})$-free} means that
no induced subposet is a  
disjoint union of an $a$-element chain and a $b$-element chain.
%(See \cite[Ch.\,3]{StanEC1}.)
Such posets are also called {\em unit interval orders}.
It is known that unit interval orders may be naturally labeled by the
numbers $[n]$ so that $i <_P j$ implies $i < j$ as integers
and so that the conditions $i < j < k$ and $i$ incomparable to $k$ in $P$
imply that $\{ i, j, k \}$ is an antichain in $P$~\cite[Sec.\,3]{DK}.
(See \cite[Prop.\,2.4]{SkanReed} for the algorithm.)
%, \cite[Sec.\,8.2]{Trott}.)
It is known that there are $\tfrac 1{n+1}\tbinom{2n}n$ unit interval 
orders on $n$ elements~\cite[Sec.\,4]{DK}.
%(See \cite[Exercise 6.29]{StanEC2}.)


%{\em $(\mathbf{3}+\mathbf{1})$-free}
%and
%{\em $(\mathbf{2}+\mathbf{2})$-free}
%where
%four-element {\em unit interval orders},
%posets containing no induced subposet isomorphic to 
%\begin{thm}
%Let $w \in \sn$ avoid the pattern $312$.  Then the poset $Q(F_w)$
%is a unit interval order.
%\end{thm}

%($\star$ Include results from chrom.tex which connect immanants, 
%unit interval orders, descending star networks, and 
%Kazhdan-Lusztig basis elements.)

%($\star$ Introduce this.)
\begin{thm}\label{t:dsnuio}
The posets 
$\{ P(F_v) \,|\, v \in \sn \text{ avoids } 312 \}$
are precisely the 
%$\tfrac 1{n+1}\tbinom{2n}n$ 
unit interval orders on $n$ elements.
\end{thm}
\begin{proof}
Let $\mathcal P$ and $\mathcal U$ be the two sets of posets in the
theorem.
% of $n$-element unit interval orders,
%and let $\mathcal P_2$ be the set 
%$\{ P(F_w) \,|\, w \in \sn \text{ avoids } 312 \}$.
%of descending star networks of order $n$.
We define a map $\zeta: \mathcal U \rightarrow \mathcal P$
as follows.
% from $n$-element unit interval orders to descending star networks of order $n$ as follows.
By the above fact on incomparability, 
%Proposition~\ref{p:} 
we may naturally label $P \in \mathcal U$ by
$1, \dotsc, n$ so that its maximal antichains are
%Let unit interval order $P$ have maximal antichains 
$[a_1, b_1], \dotsc, [a_t,b_t]$, 
with $a_1 > \cdots > a_t$.
%, $b_1 > \cdots > b_t$.
Now define $F$ to be the descending star network
corresponding to the concatenation 
$G_{[a_1,b_1]} \circ \cdots \circ G_{[a_t, b_t]}$,
and let $(\pi_1, \dotsc, \pi_n)$ be the unique path family of type $e$ that
covers $F$.  Let $\zeta(P)$ be the poset $P(F)$.

We claim that for each poset $P \in \mathcal U$, 
the map $i \mapsto \pi_i$ is an isomorphism of 
%the posets 
$P$ and $\zeta(P)$.
To see this, note that $i <_P j$ if and only if $i < j$ as integers
and $i, j$ belong to no common 
%maximal 
antichain 
%$[a_k,b_k]$ 
in $P$,  
i.e., if and only if $i < j$ and $i,j$ belong to no common interval
$[a_k,b_k]$ defining the concatenation 
$G_{[a_1,b_1]} \circ \cdots \circ G_{[a_t,b_t]}$ of star networks.
But this is true if and only if we have $\pi_i <_{\zeta(P)} \pi_j$.
It follows that the 
$\tfrac 1{n+1}\tbinom{2n}n$ images $\{ \zeta(P) \,|\, P \in \mathcal U \}$
are pairwise nonisomorphic.
On the other hand, by Theorem~\ref{t:dsn312}, we have
%\begin{equation*}
%$\mathcal F = \{ F_w \,|\, w \in \sn \text{ avoids } 312 \}$.
%Thus 
$|\mathcal P| \leq \tfrac 1{n+1}\tbinom{2n}n$. 
%\end{equation*}
%Omitted.
The claim follows.
%Let $\mathcal P$ be the set of $n$-element unit interval orders,
%and let $\mathcal F$ be the set of descending star networks of order $n$.
%By Theorem~\ref{t:dsn312}, we have
%$\mathcal F = \{ F_w \,|\, w \in \sn \text{ avoids } 312 \}$.
%Thus $|\mathcal F| \leq \tfrac 1{n+1}\tbinom{2n}n$. 
%
%We define a map $\zeta: \mathcal P \rightarrow \mathcal F$
%as follows.
%By Proposition~\ref{p:} we may naturally label $P \in \mathcal P$ by
%$1, \dotsc, n$ so that its maximal antichains are
%$[a_1, b_1], \dotsc, [a_t,b_t]$, 
%with $a_1 > \cdots > a_t$, $b_1 > \cdots > b_t$.
%Now define $\zeta(P)$ to be the descending star network
%corresponding to the concatenation 
%$G_{[a_1,b_1]} \circ \cdots \circ G_{[a_t, b_t]}$,
%and let $(\pi_1, \dotsc, \pi_n)$ be the unique path family of type $e$ that
%covers $\zeta(P)$.  
%
%We claim that for each poset $P \in \mathcal P$, 
%the map $i \mapsto \pi_i$ is an isomorphism of 
%$P$ and $P(\zeta(P))$.
%To see this, note that $i <_P j$ if and only if $i < j$ as integers
%and $i, j$ belong to no common 
%antichain 
%in $P$,  
%i.e., if and only if $i < j$ and $i,j$ belong to no common interval
%$[a_k,b_k]$ defining the concatenation 
%$G_{[a_1,b_1]} \circ \cdots \circ G_{[a_t,b_t]}$ of star networks.
%But this is true if and only if we have $\pi_i <_{P(\zeta(P))} \pi_j$.
\end{proof}

While a star network is covered by a unique path family of type $e$,
a concatenation of star networks need not be.
Nevertheless, such a concatenation is covered by a unique noncrossing path
family of type $e$: the concatenation of the unique path families of type
$e$ that cover the component star networks.  
%$G$,
%a bijective skeleton $G$ of order $n$, 
Given a concatenation $G$ of star networks, 
let $\hat P(G)$ be the $n$-element subposet of $P(G)$
induced by the paths in the unique noncrossing path family of type $e$
which covers $G$.
For example, the first two figures in (\ref{eq:cover}) show 
the two path families of type $e$ which cover $G_{[2,4]} \circ G_{[1,3]}$.  
Call these 
%$\pi = (\pi_1, \pi_2, \pi_3, \pi_4)$ 
$(\pi_1, \pi_2, \pi_3, \pi_4)$ 
and 
%$\pi' = (\pi_1, \pi'_2, \pi'_3, \pi_4)$.
$(\pi_1, \pi'_2, \pi'_3, \pi_4)$, respectively.
Ordering all six of these source-$i$-to-sink-$i$ paths, 
or only the mutually noncrossing paths $\{ \pi_1, \pi_2, \pi_3, \pi_4 \}$, 
we form the posets
%we have
%the poset $P(G_{[2,4]} \circ G_{[1,3]})$, 
%whose subposet induced by the 
%Only $\pi$ consists of
%only one of which is noncrossing.  
%These two path families and 
%Thus 
%and 
%$\hat P(G_{[2,4]} \circ G_{[1,3]})$, 
%are 
%($\star$ Draw them.)
\begin{equation*}
P(G_{[2,4]} \circ G_{[1,3]}) = 
\raisebox{-9mm}{
\includegraphics[height=18mm]{tableaux/p}}, \qquad
\hat P(G_{[2,4]} \circ G_{[1,3]}) = 
\raisebox{-9mm}{
\includegraphics[height=18mm]{tableaux/phat}}, 
\end{equation*}
respectively.

It is easy to show that 
for a concatenation $G$ of star networks,
%bijective skeletons, 
the poset $\hat P(G)$ does not
depend upon the ordering of the factors of $G$.
%leads to a concatenation the same poset $\hat P$. 
\begin{prop}\label{p:permisom}
Let $[a_1,b_1],\dotsc, [a_t,b_t]$ be 
%(not necessarily distinct) 
subintervals of $[n]$.
%$\{ G_{[a_i,b_i]} \,|\, 1 \leq i \leq t \}$ be distinct star networks
%bijective skeletons 
%of order $n$.  
Then for any permutation $u \in \mfs t$, we have
%$\hat P(G_{L_1} \circ \cdots \circ G_{L_t}) \cong
%\hat P(G_{L_{u_1}} \circ \cdots \circ G_{L_{u_t}})$.
$\hat P(G_{[a_1,b_1]} \circ \cdots \circ G_{[a_t,b_t]}) \cong
\hat P(G_{[a_{u_1},b_{u_1}]} \circ \cdots \circ G_{[a_{u_t},b_{u_t}]})$.
\end{prop}
\begin{proof}
Define $G = G_{[a_1,b_1]} \circ \cdots \circ G_{[a_t,b_t]}$,
$H = G_{[a_{u_1},b_{u_1}]} \circ \cdots \circ G_{[a_{u_t},b_{u_t}]}$,
%$H = G_{L_{u_1}} \circ \cdots \circ G_{L_{u_t}}$,
%$G = G^{(1)} \circ \cdots \circ G^{(t)}$,
%$H = G^{(u_1)} \circ \cdots \circ G^{(u_t)}$,
and let $(\rho_1, \dotsc, \rho_n)$, $(\tau_1, \dotsc, \tau_n)$
be the unique noncrossing path families of type $e$ 
covering $G$, $H$, respectively.
%$G$,
%$H$ be concatenations of star networks and 
%Let $(\rho'_1, \dotsc, \rho'_n)$
%be the unique noncrossing path family of type $e$ 
%covering $G' = G^{(u_1)} \circ \cdots \circ G^{(u_t)}$,
%and 
For $i = 1, \dotsc, t$, let $(\pi_1^{(i)}, \dotsc, \pi_n^{(i)})$ 
be the unique noncrossing path family of type $e$ covering 
$G_{[a_i,b_i]}$.
%$G^{(i)}$.
%covering these (respectively).
%Then we have 
%Let $R$ be the subposet of $P(G \circ H)$ consisting of the $n$ paths
%$(\pi_1 \circ \rho_1, \dotsc, \pi_n \circ \rho_n )$. 
%is the unique path family of type $e$ covering $G \circ H$ and we have
For $j, k \in [n]$, the definition of $\hat P(G)$ implies we have 
$\rho_j \leq_{\hat P(G)} \rho_k$ if and only if
%$\smash{\pi_j^{(i)} \leq_{\hat P(G^{(i)})} \pi_j^{(i)}}$ 
$\pi_j^{(i)} \leq_{\hat P(G_{[a_i,b_i]})} \pi_k^{(i)}$ 
for $i = 1, \dotsc, t$.
But this condition is also equivalent to 
$\smash{\tau_j \leq_{\hat P(H)} \tau_k}$.
\end{proof}

%\begin{lem}
%Let $\{ G^{(i)} \,|\, 1 \leq i \leq t \}$ be star networks,
%let $(\rho_1, \dotsc, \rho_n)$
%be the unique noncrossing path family of type $e$ 
%covering $G = G^{(1)} \circ \cdots \circ G^{(t)}$,
%and for $i = 1, \dotsc, t$, let $(\pi_1^{(i)}, \dotsc, \pi_n^{(i)})$ 
%be the unique path family of type $e$ covering $G^{(i)}$.
%Then we have
%$\rho_j \leq_{\hat P(G)} \rho_k$ if and only if
%$\pi_j^{(i)} \leq_{P(G^{(i)})} \pi_j^{(i)}$ for $i = 1, \dotsc, t$.
%\end{lem}
%\begin{proof}
%Omitted.
%\end{proof}

%($\star$ Introduce this.)
It is also easy to show that the path poset of a zig-zag network $F$
may be obtained directly from the concatenation of star networks 
which lead to $F$ as in Section~\ref{s:dsn}.
\begin{prop}\label{p:hatisom}
Let $F$ be a zig-zag network constructed from a concatenation $G$ of
star networks as after (\ref{eq:concat}).
Then $P(F)$ is isomorphic to $\hat P(G)$.
\end{prop}
\begin{proof}
Let $(\rho_1, \cdots, \rho_n)$ and $(\tau_1,\dotsc,\tau_n)$ be the unique
noncrossing path families of type $e$ in $G$ and $F$, respectively,
and let $x_1,\dotsc,x_t$ be the vertices of $F$ which correspond to the
central vertices of the $t$ star networks in (\ref{eq:concat}).
For $i < j$, we have $\rho_i <_{\hat P(G)} \rho_j$ if and only if $\rho_i$, 
$\rho_j$ share none of the central vertices of the $t$ star networks.
But this condition holds if and only if $\tau_i$, $\tau_j$ share none of
the vertices $x_1, \dotsc, x_t$.
\end{proof}

By Propositions~\ref{p:permisom} and \ref{p:hatisom},
it is possible to have an isomorphism of path posets for nonisomorphic 
zig-zag networks.  Define an equivalence relation $\sim$ on \pavoiding
permutations by 
\begin{equation}\label{eq:posetequiv}
v \sim w \quad \text{ if and only if } \quad P(F_v) \cong P(F_w).
\end{equation}
For example, it is easy to see the equivalence of the four permutations 
corresponding to the 
seventh descending star network in (\ref{eq:xfigures2}) 
and the fourth, seventh, and eighth zig-zag networks in (\ref{eq:xfigureszz}):
in each case the path poset is isomorphic to the seventh unit interval order in
(\ref{eq:uio}).
It is also easy to see that we have $w \sim w^{-1}$ for $w$ \avoidingp:
the networks $F_w$ and $F_{w^{-1}}$ differ only by reflection in a vertical line. 
 
%($\star$ Introduce this.)
\begin{thm}\label{t:pfvpfu}
Each equivalence class of the relation $\sim$ (\ref{eq:posetequiv})
contains exactly one representative which avoids the pattern $312$.
%Let $w \in \sn$ \avoidp.  Then there exists a $312$-avoiding permutation $v$
%which satisfies $P(F_w) \cong P(F_v)$.
\end{thm}
\begin{proof}
Fix $w \in \sn$ \avoidingp, let 
$G = G_{[c_1,d_1]} \circ \cdots \cdots \circ G_{[c_t,d_t]}$ 
be the concatenation of star networks which leads to the zig-zag network 
$F_w$ as after (\ref{eq:xfigureszz}),
and let $u \in \mfs t$ be the unique permutation
%$[c_{u_1},d_{u_1}], \dotsc, [c_{u_t},d_{u_t}]$ be the unique rearrangement
%of the intervals 
satisfying $c_{u_1} > \cdots > c_{u_t}$.
Then the concatenation 
$G_{[c_{u_1},d_{u_1}]} \circ \cdots \cdots \circ G_{[c_{u_t},d_{u_t}]}$ leads to
a descending star network $F_v$.  
By Theorem~\ref{t:dsn312}, $v$ avoids the pattern $312$,
and by Propositions~\ref{p:permisom} and \ref{p:hatisom} we have
$P(F_v) \cong P(F_w)$.
By Theorem~\ref{t:dsnuio}, $v$ is the only $312$-avoiding permutation in
its equivalence class.
\end{proof}

\ssec{Planar network tableaux}
% and interpretation of $\sn$-class functions}
To combinatorially interpret evaluations of 
$\sn$-class functions and $\hnq$-traces,
we will repeatedly fill a (French) Young diagram 
with a path family
$\pi = (\pi_1,\dotsc,\pi_n)$ 
covering a zig-zag network
%descending star network 
$F_w$,
%which come from a bijective skeleton $F$,
and will call the resulting structures {\em $F_w$-tableaux},
or more specifically {\em $\pi$-tableaux}.
(See, e.g., \cite[Sec.\,2]{RemTrans} for French notation.)
%If an $F_w$-tableau $U$ contains a path family $\pi$ of type $v$,
If 
%a fixed path family 
$\pi$ 
%in $F_w$ 
has type $v$,
then we will also say that each $\pi$-tableau has {\em type $v$.}
Since $\pi$ 
%a path family $\pi = (\pi_1, \dotsc, \pi_n)$ in a planar network $F_w$ 
induces a subposet $Q_\pi$ of the poset $Q(F_w)$,
%these 
$\pi$-tableaux form a special case of Gessel and Viennot's 
%$F_w$-tableaux are a special case of Gessel and Viennot's 
%{\em $P$-arrays}~\cite{GV}), commonly called 
{\em $P$-tableaux}~\cite{GV}: they are
%.
%According to this common terminology, 
%if $Q_\pi$ is the subposet of $Q(F_w)$
%induced by the elements $\pi_1, \dotsc, \pi_n$ of a path family $\pi$,
%then 
%a $\pi$-tableau
%an $F_w$-tableau containing $\pi$ 
%is a 
$Q_\pi$-tableaux.

Several properties which $\pi$-tableaux may posess can be defined 
for $P$-tableaux where $P$ is 
an arbitrary poset.
%any {\em naturally labeled} 
%poset on the set $\{\pi_1, \dotsc, \pi_n\}$, 
%i.e., $\pi_i <_P \pi_j$ implies $i < j$ as integers.  
%Let $U$ be a $P$-tableau.
%For any poset $P$, 
We say that a $P$-tableau $U$ has {\em shape $\lambda$} 
for some partition $\lambda = (\lambda_1,\dotsc,\lambda_r)$ if it
has $\lambda_i$ cells in row $i$ for all $i$.
If $U$ has $\lambda_i$ cells in {\em column $i$} for all $i$, 
we say that $U$ has shape $\lambda^\tr$, 
where 
%In this case 
we define $\lambda^\tr$ 
to be the partition whose $i$th part is equal to
the number of cells in row $i$ of $U$.
Call an element $x \in P$
%a path $\pi_i$ 
a {\em nontrivial record} in a row of $U$
%a $P$-tableau
if it is greater in $P$ than all entries
%paths 
appearing to its left in the same row,
and if it is not the leftmost entry of its row.
%\begin{enumerate}
\begin{itemize}
\item Call $U$
%a $P$-tableau
%An English (French) $P$-tableau is called 
{\em column-strict} ({\em row-strict}\,)
%if whenever paths $\pi_i, \pi_j$ appear consecutively from
if whenever elements $x$, $y$ appear consecutively from
%if whenever elements $i_1, \dotsc, i_r$ appear from 
%top to bottom 
bottom to top in a column (left to right in a row), then we have 
%that
$x <_P y$.
%$\pi_i <_P \pi_j$.
%$i_1 <_P \cdots <_P i_r$.
% is a chain in $P$.
\item Call $U$
%a $P$-tableau
%An English or French $P$-tableau is called 
{\em row-semistrict}
if whenever elements $x$, $y$
%$\pi_i, \pi_j$ 
appear consecutively from left to right in a row,
%then we have $\pi_i <_P \pi_j$ or $\pi_i$ incomparable to $\pi_j$ in $P$.
then we have $x <_P y$ or $x$ incomparable to $y$ in $P$.
\item Call $U$
%a row-semistrict $P$-tableau 
{\em cyclically row-semistrict}
if it is row-semistrict and the above condition 
%on $i_1$ and $i_2$ 
also holds
%An English or French $P$-tableau is called {\em row-cyclic} if 
%whenever $i_1, i_2$ appear consecutively in a row, or 
when $x$, $y$
%$\pi_i, \pi_j$ 
are the rightmost and leftmost (respectively) 
entries in a row.
% and $i_2$ is first in that row.
%then we must have $i_1 <_P i_2$ or $i_1$ incomparable to $i_2$ in $P$.
\item Call $U$
%a $P$-tableau 
{\em standard}
if it is both column-strict and row-semistrict.
\item Call $U$
%a $P$-tableau 
{\em record-free} 
if no row contains a nontrivial record.
%\end{enumerate}
\end{itemize}
Another property of $\pi$-tableaux depends upon each path being labeled by
its source vertex.
Call a row of $U$
%a $P$-tableau 
{\em left anchored} ({\em right anchored}\,)
if its leftmost (rightmost) entry has the least 
%label
source vertex
%index 
of all paths in the row.
%\begin{enumerate}[(6)]
\begin{itemize}%[(6)]
\item Call $U$
%a $P$-tableau 
{\em left-anchored} ({\em right-anchored}\,) 
if each row is left-anchored (right-anchored).
%Call an $F_w$-tableau {\em record-free} if no row contains a nontrivial record,
%and {\em left-anchored} ({\em right-anchored}) 
%if each row is left-anchored (right-anchored).
%\end{enumerate}
\end{itemize}
%Call paths $(\pi_i,\pi_j)$ an {\em inversion} in $U$
%a $P$-tableau 
%if the paths are incomparable in $P$ with $i < j$ 
%and $\pi_j$ appearing in an earlier column than $\pi_i$.
%Let $\inv(U)$ denote the number of inversions in 
%a $P$-tableau 
%$U$.
%Several properties of $P$-tableaux have appeared in the literature.
%possess are easily stated in 
%terms of the poset $Q_\pi$.
%We say that an $F_w$-tableau $U$ has {\em shape $\lambda$} 
%for some partition $\lambda = (\lambda_1,\dotsc,\lambda_r)$ if it
%has $\lambda_i$ cells in row $i$ for all $i$.
%If $U$ has $\lambda_i$ cells in {\em column $i$} for all $i$, 
%we say that $U$ has shape $\lambda^\tr$. 
%In this case we define $\lambda^\tr$ 
%to be the partition whose $i$th part is equal to
%the number of cells in row $i$ of $U$.
%where $\lambda^\tr = (\lambda_1^\tr, \dotsc, \lambda_{\lambda_1}^\tr)$.
%Given an $F$-tableau $U$ containing a path family 
%$(\pi_1, \dotsc, \pi_n)$,
%($\star$ Define a few tableaux terms.)
%Suppose that a row of an $F_w$-tableau contains
%the paths $(\pi_{i_1}, \dotsc, \pi_{i_k})$.
% element appears 
%Call path $\pi_{i_j}$ a {\em nontrivial record}
%in the row if it lies strictly above $\pi_{i_1}, \dotsc, \pi_{i_{j-1}}$ in $F_w%$,
%and if $j > 1$.
%Call the row {\em left-anchored} if $i_1 = \min \{i_1, \dotsc, i_k \}$, and
%{\em right-anchored} if $i_k = \min \{i_1, \dotsc, i_k \}$.
%Trivially, $\pi_{i_1}$ is a record.
%We define several properties of an $F$-tableau in terms of the poset $Q$
More
%Other 
properties of $\pi$-tableaux depend upon the fact that
each element of the poset $Q_\pi$ has a source vertex and a 
(potentially different) sink vertex.
Given a $\pi$-tableau $U$,  
%an $F_w$ tableaux $U$ containing a path family $\pi = (\pi_1, \dotsc, \pi_n)$, 
let $L(U)$ and $R(U)$ denote the Young tableaux of integers
obtained from $U$ by replacing paths $\pi_1,\dotsc,\pi_n$
with their corresponding source and sink indices, respectively.
%Several properties of a $\pi$-tableau are not easily stated in 
%terms of a poset of the form $Q_\pi$.
%in terms of the poset $Q$
%and the tableaux $L(U)$ and $R(U)$.
%Additionally,
%Let $U$ be an $F$-tableau of type $w$.  
%call $U$
%\begin{enumerate}
\begin{itemize}
%\item Call $U$ {\em column-strict} if
%whenever paths $\pi_{i_1}, \dotsc, \pi_{i_r}$ appear from 
%bottom to top in a column, then we have 
%$\pi_{i_1} <_Q \cdots <_Q \pi_{i_r}$.
%\item Call $U$ {\em row-semistrict} if
%whenever paths $\pi_{i_1}$, $\pi_{i_2}$ 
%appear consecutively (from left to right) in a row, we have
%$\pi_{i_1} <_Q \pi_{i_2}$ or $\pi_{i_1}$ is incomparable to $\pi_{i_2}$ in $Q$.
%\item Call $U$ {\em cyclically row-semistrict}
%if it is row-semistrict and 
%the condition above applies also
%to paths $\pi_{i_1}$, $\pi_{i_2}$ appearing last and first (respectively) 
%in the same row.
%\item Call $U$ {\em standard} if it is column-strict and row-semistrict.
\item 
Call $U$ {\em row-closed} 
%if $L(U)$ is row-strict 
%(entries increase from left to right) and 
if each row of $R(U)$ is a permutation of the corresponding row of $L(U)$.
%\item
\item 
Call $U$ {\em left row-strict} 
if $L(U)$ is row-strict as a $\mathbb Z$-tableau. 
%(entries increase from left to right) and 
%if each row of $R(U)$ is a permutation of the corresponding row of $L(U)$.
%\item
\item Call $U$ {\em cylindrical} if
for each row of $L(U)$ containing indices
$i_1, \dotsc, i_k$ from left to right, 
% paths
%$\pi_{i_1}, \pi_{i_2}, \dotsc, \pi_{i_{k-1}}, \pi_{i_k}$ 
the corresponding row of $R(U)$ contains
%$\pi_{i_1}, \dotsc, \pi_{i_k}$ from left to right, 
%these $k$ paths terminate at sinks 
$i_2, \dotsc, i_k, i_1$ from left to right.
%, respectively.  
%($\star$ Include Wachs-Shareshian minimum-free condition.)
%\end{enumerate}
\end{itemize}

It will be convenient to let 
$\mathcal T(F_w,\lambda)$ 
be the set of $F_w$-tableaux of shape $\lambda$.

\begin{lem}\label{l:dropparens}
Let $v \in \sn$ avoid the pattern $312$, and fix $\lambda \vdash n$.
%and fix a row-strict Young tableau $T$ of shape $\lambda$. 
%Then 
%tableaux in 
Within the set $\mathcal T(F_v,\lambda)$, tableaux 
which are row-closed and left row-strict
correspond bijectively to 
%tableaux in $\mathcal T(F_w,\lambda)$ 
tableaux which are row-semistrict of type $e$.
%we have
%\begin{equation*}
%\item[$(ii$-$a)$] $\eta^\lambda(C'_w(1)) = 
%\# \{ U \in \mathcal T(F_w,\lambda) \,|\, 
%U \text{ row-closed, left row-strict } \}
%%\item[$(ii$-$b)$] $\eta^\lambda(C'_w(1)) = 
%= \# \{ U \in \mathcal T(F_w,\lambda) \,|\, 
%U \text{ row-semistrict of type } e \}.
%\end{equation*}
\end{lem}
\begin{proof}
Observe that for a row-closed, left row-strict tableau
$V \in \mathcal T(F_v,\lambda)$, 
the tableau $V_k$ is itself a row-closed, left row-strict tableau
in $\mathcal T(F_v|_{I_k}, \lambda_k)$, where $I_k$ is the set of indices
appearing in $L(V_k)$.
Similarly, for a row-semistrict tableau $V' \in \mathcal T(F_v,\lambda)$
of type $e$, the tableau $V'_k$ is itself a row-semistrict tableau of type $e$ 
in $\mathcal T(F_v|_{I_k}, \lambda_k)$.
Letting $r = \lambda_k$, we have that
$F_v|_{I_k}$ is a descending star network of the form $F_u$
for some $u \in \mfs r$ avoiding the pattern $312$.
%$u \in \mfs {\lambda_k}$,
We therefore construct a bijection which preserves the 
set of path indices appearing in each row of a tableau,
and we state it as a product of bijections on one-rowed tableaux.
%Fix $r \leq n$, and let $v \in \mfs r$ avoid the pattern $312$.
%We define a bijection 

%Define a 
Map each  
left row-strict tableau $U \in \mathcal T(F_u,(r))$ 
to a row-semistrict tableau $U'$ of type $e$ 
in $\mathcal T(F_u,(r))$ as follows.
Let $\rho = (\rho_1, \dotsc, \rho_r)$ be the unique path family of 
type $e$ covering $F_u$.
Let 
%$U \in \mathcal T(F_v,(r))$ 
$U$ contain
%be a left row-strict tableau containing 
the path family $\pi = (\pi_1, \dotsc, \pi_r)$ from left to right, 
and define $w \in \mfs r$ to be the word of right indices of these paths.
%with
%$R(U)$ containing the indices $(w_1, \dotsc, w_r)$ from left to right,
%for some $w \in \mfs r$.
Write $w^{-1}$ in cycle notation, with each cycle starting with its 
greatest element, and cycles ordered by increasing greatest elements.
(See Cycle Structure subsection in \cite[Sec.\,1.3]{StanEC1}).  
Drop the parentheses, and interpret the resulting
string $x = x_1 \cdots x_r$ of letters as the one-line notation of an
element of $\mfs r$.  Then write the paths $\rho_{x_1}, \dotsc, \rho_{x_r}$
from left to right in $U'$.

To see that $U'$ is row-semistrict, assume that we have 
$\rho_{x_i} >_{P(F_u)} \rho_{x_{i+1}}$ for some $i$.  Then there is no
path from source $x_{i+1}$ to sink $x_i$ in $F_u$.
If $x_i$ and $x_{i+1}$ belong to the same cycle of $w^{-1}$,
then $w_{x_{i+1}} = x_i$ and there is a path from $x_{i+1}$ to $x_i$ in $F_u$,
a contradiction. 
If $x_i$ and $x_{i+1}$ do not belong to the same cycle of $w^{-1}$,
then $x_{i+1} > x_i$ as integers, contradicting the assumed 
inequality in $P(F_u)$.

To see that the map $U \mapsto U'$ is a bijection, we construct its inverse.
Let $V$ be a row-semistrict tableau of type $e$ in $\mathcal T(F_u,(r))$, 
containing paths $\rho_{x_1}, \dotsc, \rho_{x_r}$ from left to right. 
Define $w \in \sn$ to be the permutation whose cycle notation is given by
\begin{equation*}
(x_1, \dotsc, x_{i_1 - 1})(x_{i_1},\dotsc,x_{i_2 - 1}) \cdots (x_{i_k},\dotsc,x_r),
\end{equation*}
%, let $w \in \mfs r$ by the permutation whose cycle
%notation is given by
where $x_1, x_{i_1}, x_{i_2}, \dotsc, x_{i_k}$ are the 
%positions of nontrivial 
records of the word
$x_1 \cdots x_r$, i.e., $x_{i_j} = \max \{ x_1,\dotsc,x_{i_j}\ntnsp \}$.
Then write $w^{-1} = w^{-1}_1 \cdots w^{-1}_r$ in one-line notation and
define $V'$ to be the tableau in $\mathcal T(F_u,(r))$ 
whose $i$th 
entry is the unique path in $F_u$ from source $i$ to sink $w^{-1}_i$.  
It is clear that this map, if well defined, 
is inverse to the map $U \mapsto U'$, 
and therefore that the two are bijections. 
(See \cite[Sec.\,1.3]{StanEC1}).  

To see that the necessary paths exist in $F_u$,
%such a path exists for each index $i$, 
consider a cycle $(x_j, \dotsc, x_{j+\ell})$ of $w$ and the pairs 
$(i,w^{-1}_i) \in 
\{(x_{j+a+1}, x_{j+a})\,|\, 1 < a \leq \ell \} \cup \{(x_{j+\ell}, x_j)\}$.
Since $V$ is row-semistrict, i.e.,
$\rho_{x_{j+a}} \not >_{P(F_u)} \rho_{x_{j+a+1}}$,
any integer inequality $x_{j+a} > x_{j+a+1}$ 
implies that there are paths in $F_u$ from sources $x_{j+a}$ and $x_{j+a+1}$ 
to (both) sinks $x_{j+a}$ and $x_{j+a+1}$.
In particular, there are paths in $F_u$ from sources $x_j$ and $x_{j+1}$
to sinks $x_j$ and $x_{j+1}$.  
Now assume that there are paths from 
source $x_{j+s}$ to sink $x_{j+s-1}$ and from source $x_j$ to sink $x_{j+s}$,
and consider 
%Now assume that for $a = 1, \dotsc, s$, there are paths from 
%source $x_{j+a}$ to sink $x_{j+a-1}$ and from source $x_j$ to sink $x_{j+a}$.  
%Consider 
the pair $(x_{j+s+1}, x_{j+s})$.  
If $x_{j+s+1} < x_{j+s}$, then by the above argument there is a path
from source $x_{j+s+1}$ to sink $x_{j+s}$.  
Since there are paths from sources $x_j$ and $x_{j+s+1}$ to sink $x_{j+s}$,
the two sources must belong to the same connected component of $F_u$.
By the comment following Observation~\ref{o:sourcetosink}, we also
have a path from source $x_j$ to sink $x_{j+s+1}$.  
If on the other hand $x_{j+s+1} < x_{j+s}$, 
then since $x_j$ is the maximum index in its cycle
we have $x_j > x_{j+s+1} > x_{j+s}$.  Since there are paths in $F_u$ from
source $x_j$ to sink $x_{j+s}$ and from source $x_{j+s+1}$ to sink $x_{j+s+1}$,
Observation~\ref{o:pnetintersect}
implies that there are also a paths from source
$x_j$ to sink $x_{j+s+1}$ and from source $x_{j+s+1}$ to $x_{j+s}$.
%($\star$ Why is there also a path from source $x_j$ to sink $x_{j+s+1}$?)  
By induction, we have that for $a = 1, \dotsc, \ell$,
there is a path from source $x_{j+a}$ to sink $x_{j+a-1}$,
and that there is also a path from source $x_j$ to sink $x_{j+\ell}$.
\end{proof}

\ssec{Interpretation of $\sn$-class function evaluations}
The equivalence relation in (\ref{eq:posetequiv}) has applications in
the enumeration of certain $F$-tableaux and in the evaluation of $\sn$-class
functions.

\begin{thm}\label{t:thetavw}
Let $v, w \in \sn $ \avoidp\ and satisfy $v \sim w$,
and let $X$ be a property of $F$-tableaux 
which depends only upon the poset $P(F)$ (rather than on $Q(F)$).
Then $F_v$-tableaux and $F_w$-tableaux having property $X$ are in
bijective correspondence.  Moreover, for any $\sn$-class function $\theta$
we have $\theta(C'_v(1)) = \theta(C'_w(1))$.
%For $v, w \in \sn $ \avoidingp\ and satisfying $v \sim w$,
%we have
%$\theta(C'_w(1)) = \theta(C'_v(1))$.
\end{thm}
\begin{proof}
%Observe that 
%for any property $X$ of $F_w$-tableaux which depends 
%only upon the poset $P(F_w)$,
Since the property $X$ depends only upon the poset $P(F_v) \cong P(F_w)$,
we have a bijection between the sets of 
$F_v$-tableaux 
%which depends only upon the poset is also a 
and $F_w$-tableaux having property $X$.
%It follows that for such a property $X$, any identity of the form
%%\begin{equation}\label{eq:property}
%$\theta(C'_v(1)) =
%\# \{ U \,|\, U \text{ an $F_v$-tableau with property } X \}$
%%\end{equation}
%implies that we have
%\begin{equation}\label{eq:holdsforpavoidingtoo}
%\begin{aligned}
%\theta(C'_w(1)) =
%\# \{ U \,|\, U \text{ an $F_w$-tableau with property } X \}.
%\end{aligned}
%\end{equation}
%In particular, 
%Statement 

Now apply Lindstr\"om's Lemma and (\ref{eq:thetaCimmB}) to
the first Littlewood-Merris-Watkins identity in (\ref{eq:inducedimmssubmat})
to see that
%obtain
%(i) 
%of the theorem
%implies that we have
for all $\lambda \vdash n$, the evaluations
$\epsilon^\lambda(C'_v(1))$
and $\epsilon^\lambda(C'_w(1))$
% = \epsilon^\lambda(C'_v(1))$
are equal to the numbers of 
column-strict $F_v$-tableaux and $F_w$-tableaux, respectively,
of type $e$ and shape $\lambda$.
%column-strict $F_w$-tableaux of type $e$, respectively.
%This in turn 
%is equal to the number of column-strict $F_v$-tableaux
%of type $e$, i.e., 
Since column-strictness of these tableaux depends only upon 
$P(F_v) \cong P(F_w)$,
the above bijection implies that we have 
$\epsilon^\lambda(C'_v(1)) = \epsilon^\lambda(C'_w(1))$ for all $\lambda \vdash n$.
Since $\{ \epsilon^\lambda \,|\, \lambda \vdash n \}$ is a basis
for the space of $\sn$-class functions, each class function
%element 
$\theta$ 
%of this space 
satisfies
%we have the equality
$\theta(C'_v(1)) = \theta(C'_w(1))$.
% as well.
%desired result.
\end{proof}
%Let $\theta$ be an $\sn$-class function.
%If the posets $P(F_v)$, $P(F_w)$ of two zig-zag networks are dual,
%rather than isomorphic, we still have
%$\theta(C'_v(1)) = \theta(C'_w(1))$.  
%In this case $v$ and $w$ satisfy $v \sim w_0ww_0$, where
%$w_0$ is the longest element of $\sn$, 
%and the equations
%\begin{equation}\label{eq:dualposeteq}
%\theta(C'_{w_0ww_0}(1)) = \theta(w_0 C'_w(1) w_0) = \theta(C'_w(1))
%\end{equation}
%hold for all $w \in \sn$.
%It is possible to have 
%$v \not \sim w$.
%It is easy to see that a zig-zag network $F_w$ and its reflection
%in a vertical line ($F_{w^{-1}}$) are isomorphic.  Thus we have
%$w \sim w^{-1}$ and $\theta(C'_w(1)) = \theta(C'_{w^{-1}}(1))$.
%On the other hand, $F_w$ is not so related to its reflection in a horizontal
%Indeed, let 
%Then 
%$F_w$ and its reflection in a horizontal
%line ($F_{w_0ww_0}$, where $w_0$ is the longest element of $\sn$)
% has one-line notation $n(n-1) \cdots 1$):
%the posets $P(F_w)$, $P(F_{w_0ww_0})$ are dual, rather than isomorphic.
%Nevertheless we have for all $w \in \sn$ that
%While pairs $(w, w_0ww_0)$ do not in general satisfy $w \sim w_0ww_0$
%(the posets $P(F_{w_0ww_0})$ and $P(F_w)$ are dual,
%rather than isomorphic),
%it is easy to see that we also have 
%\begin{equation}\label{eq:dualposeteq}
%\theta(C'_{w_0ww_0}(1)) = \theta(C'_w(1)),
%\end{equation}
%where $w_0$ is the ``longest'' permutation in $\sn$, having one-line notation
%$n(n-1) \cdots 1$.
%it is easy to show that 
%since
%e have 
%$w_0 = w_0^{-1}$, we have
% and therefore
%\begin{equation}
%Specifically, both expressions are equal to
%$\theta(C'_{w_0ww_0}(1)) = \theta(w_0 C'_w(1) w_0) = \theta(C'_w(1))$.
%\end{equation}
%since $w_0 = w_0^{-1}$.
%also have 
%$\theta = \sum_{\lambda} c_\lambda \epsilon^\lambda$
%$\theta(C'_w(1)) = \theta(C'_v(1))$
%for any $\sn$-class function $\theta$.

For some $\sn$-class functions $\theta$, and all 
%$312$-avoiding
%$3412$-avoiding, $4231$-avoiding 
\pavoiding permutations $w$, we may combinatorially
interpret $\theta(C'_w(1))$
in terms of a zig-zag network $F_w$ as follows.
%, for $F$ a bijective skeleton, as follows.
%have the following interpretations of class functions
%($\star$ Does this hold more generally for \pavoiding permutations?)
(See \cite[p.\,288]{StanEC2} 
for information on the majorization order, used in (v-a) below.)
%Define dominance order earlier without symbol.)  
%Change the symbol $\prec$ in the interval poset to $\triangleleft$.)

%($\star$ Put $\epsilon$ first.  In proof, conclude that proving a result
%for $\pi$ tableaux of type $e$ for $312$-avoiding permutations 
%suffices for proving the result for \pavoiding permutations.
%Switch order somewhat in proposition and proof.)


\begin{thm}\label{t:sncfinterp}
%Let $F$ be a bijective skeleton.
Let $w \in \sn $ \avoidp, and
%let $v \sim w$ avoid the pattern $312$, and 
fix $\lambda = (\lambda_1, \dotsc, \lambda_r) \vdash n$.
Then we have the following.
%avoid the pattern $312$,
%patterns $3412$ and $4231$,
%and let $F_w$ be the corresponding descending star network.
\begin{enumerate}
%[$(i)$]
\item[$(i)$] $\epsilon^\lambda(C'_w(1)) = 
\# \{ U \in \mathcal T(F_w,\lambda^\tr\,) \,|\, 
U \text{ column-strict of type } e \}$.
%\item $\epsilon^\lambda(C'_w(1))$ equals the number of 
%column-strict $F_w$-tableaux of type $e$ and shape $\lambda^\tr$.
%\item[$(ii)$] For any $\sn$-class function $\theta$ we have 
%$\theta(C'_w(1)) = \theta(C'_v(1))$.
\item[$(ii$-$a)$] $\eta^\lambda(C'_w(1)) = 
\# \{ U \in \mathcal T(F_w,\lambda) \,|\, 
U \text{ row-closed, left row-strict}\, \}$.
\item[$(ii$-$b)$] $\eta^\lambda(C'_w(1)) = 
\# \{ U \in \mathcal T(F_w,\lambda) \,|\, 
U \text{ row-semistrict of type } e \}$.
%row-semistrict $F_w$-tableaux of type $e$ and shape $\lambda$.
%\item $\eta^\lambda(C'_w(1))$ equals 
%the number of row-closed, left row-strict 
%$F_w$-tableaux of shape $\lambda$.
%It also equals the number of
%row-semistrict $F_w$-tableaux of type $e$ and shape $\lambda$.
\item[$(iii)$] $\chi^\lambda(C'_w(1)) = 
\# \{ U \in \mathcal T(F_w,\lambda) \,|\, 
U \text{ standard of type } e \}$.
%\item $\chi^\lambda(C'_w(1))$ equals the number of 
%standard $F_w$-tableaux of type $e$ and shape $\lambda$.
\item[$(iv$-$a)$] $\psi^\lambda(C'_w(1)) = 
\# \{ U \in \mathcal T(F_w,\lambda) \,|\, 
U \text{ cylindrical}\, \}$.
%\item $\psi^\lambda(C'_w(1)) = 
%$ equals 
%the number of cylindrical
%$F_w$-tableaux of shape $\lambda$.  
\item[$(iv$-$b)$] $\psi^\lambda(C'_w(1)) = 
\# \{ U \in \mathcal T(F_w,\lambda) \,|\, 
U \text{ cyclically row-semistrict of type } e \}$.
%It also equals 
%the number of cyclically row-semistrict $F_w$-tableaux 
%of type $e$ and shape $\lambda$.
\item[$(iv$-$c)$] $\psi^\lambda(C'_w(1)) = 
\# \{ U \in \mathcal T(F_w,\lambda) \,|\, 
U \text{ record-free, row-semistrict of type } e \}$.
\item[$(iv$-$d)$] $\psi^\lambda(C'_w(1)) = 
\lambda_1 \ntnsp \cdots \ntnsp \lambda_r \cdot
\# \{ U \in \mathcal T(F_w,\lambda) \,|\, 
U \text{ right-anchored, row-semistrict of type } e \}$.
%($\star$ Prove and include Wachs-Shareshian conjectured interpretations?
%Stanley's M\"obius function interpretation?)
%\end{enumerate}
%\end{prop}
%\begin{question}
%Is $\psi^\lambda(\beta(F))$ equal to (something like) 
%\end{question}
\item[$(v$-$a)$] Suppose $\lambda_1 \leq 2$. We have
$\phi^\lambda(C'_w(1)) = 
\# \{ U \in \mathcal T(F_w,\lambda) \,|\, 
U \text{ column-strict of type } e \}$
if for all $\mu$ majorized by $\lambda$ we have
$\mathcal T(F_w,\mu) = \emptyset$;
%with 
otherwise we have $\phi^\lambda(C'_w(1)) = 0$.
%\item[$(v$-$a)$] For $\lambda_1 \leq 2$ we have
%\begin{equation*}
%\phi^\lambda(C'_w(1)) = 
%\begin{cases}
%0 &\text{if there exists $\mu \prec \lambda$ with 
%$T(F_w,\lambda) \neq \emptyset$},\\ 
%\# \{ U \in \mathcal T(F_w,\lambda) \,|\, 
%U \text{ column-strict of type } e \} &\text{otherwise}.
%\end{cases}
%\end{equation*}
%\item For $\lambda_1 \leq 2$, $\phi^\lambda(C'_w(1))$ equals zero
%if there exists a column-strict $F_w$-tableaux of type $e$ and 
%shape $\mu \prec \lambda$;
%otherwise it equals the number of column-strict $F_w$-tableaux 
%of type $e$ and shape $\lambda$.
\item[$(v$-$b)$] For $\lambda = (k)^r$, we have 
$\phi^\lambda(C'_w(1)) = 
\# \{ U \in \mathcal T(F_w,\lambda) \,|\, 
U \text{ column-strict, cylindrical}\, \}$.
%\item For $\lambda = k^r$, $\phi^\lambda(C'_w(1))$ equals 
%the number of column-strict cylindrical $F_w$-tableaux of shape $r^k$.
%if there exists a column-strict $F$-tableaux of shape $\mu \prec \lambda$;
%otherwise it equals the number of column-strict $F$-tableaux of shape $\lambda$.
\end{enumerate}
\end{thm}
%($\star$ Include left-anchored versions of parts (iv-a), (iv-b) of the theorem.
%Call them iv-a', iv-b' ? Rename part iv-d as iv-c'?)
\begin{proof} 
%Omitted.
%(Idea.)
%For $w$ avoiding $312$, the path matrix $B = (b_{i,j})$ 
%of $F_w$ 
%as in (\ref{eq:pathmatrix}).
%Then it is straightforward to show that we have
%satisfies
%\begin{equation}\label{eq:immcharformula}
%$\theta(C'_w(1)) = \imm{\theta}(B)$.
%\end{equation}
%(Idea.)
%For $w$ avoiding $312$, define the path matrix $B = (b_{i,j})$ 
%of $F_w$ as in (\ref{eq:pathmatrix}).
%Then it is straightforward to show that we have
%\begin{equation}\label{eq:immcharformula}
%\theta(C'_w(1)) = \imm{\theta}(B).
%\end{equation}
%
%Formulas for
%$\imm{\eta^\lambda}(x)$, 
%$\imm{\psi^\lambda}(x)$
%in Section~\ref{s:imm} and some consideration of the structure of $F_w$
%lead to the claimed interpretations of 
%$\eta^\lambda(C'_w(1))$, $\psi^\lambda(C'_w(1))$
%The formulas for $\imm{\epsilon^\lambda}(x)$ and $\imm{\phi^{k^r}}(x)$ 
%in Section~\ref{s:imm}, combined with Lindstr\"om's Lemma
%lead to the claimed interpretations of 
%$\epsilon^\lambda(C'_w(1))$, $\phi^{k^r}(C'_w(1))$.
%The formula for $\chi^\lambda(C'_w(1))$, due to Gasharov~\cite{GashInc},
%uses the fact that $P(F_w)$ is a unit interval order.
%The formula for $\phi^\lambda(C'_w(1))$ was proved by the authors in \cite{CSSkanPathTab}.
%
(i) See the proof of Theorem~\ref{t:thetavw}.
%Apply Lindstr\"om's Lemma and (\ref{eq:thetaCimmB}) to
%the first Littlewood-Merris-Watkins identity in (\ref{eq:inducedimmssubmat}).  

%(ii)
%To proceed, let $v$ be the unique $312$-avoiding permutation
%satisfying $v \sim w$
%and 

(ii-a) 
Apply the definition (\ref{eq:pathmatrix}) of path matrix
%apply Lindstr\"om's Lemma 
and (\ref{eq:thetaCimmB}) to 
the second Littlewood-Merris-Watkins identity in (\ref{eq:inducedimmssubmat}).  


%(ii) To obtain the first interpretation, 
(ii-b) Since row-semistrictness in $F_w$-tableaux of type $e$
is a property of the poset 
%$P(F_w) \cong P(F_v)$,
$P(F_w)$,
we may apply 
%(\ref{eq:holdsforpavoidingtoo}) 
Theorem~\ref{t:thetavw} and
Lemma~\ref{l:dropparens} to
%the argument preceeding 
the interpretation in (ii-a).
%it suffices to
%and simply state a bijection from 
%tableaux in $\mathcal T(F_v,\lambda)$ which are
%row-closed and left row-strict to
%tableaux in $\mathcal T(F_v,\lambda)$ which are
%row-semistrict of type $e$. 
%Fix $U$ in the first set
%containing the path family $\pi = (\pi_1, \dotsc, \pi_n)$, and let 
%the $k$th rows of $L(U)$ and $R(U)$ contain indices 
%$i_1, \dotsc, i_{\lambda_k}$ and $j_1, \dotsc, j_{\lambda_k}$ (respectively)
%from left to right, so $i_1 < \cdots < i_k$ and 
%$\{i_1,\dotsc, i_{\lambda_k} \} = \{j_1,\dotsc, j_{\lambda_k} \}$.
%Let $\rho = (\rho_1, \dotsc, \rho_n)$ be the unique path family of type $e$
%covering $F_v$, and create a row-semistrict tableau $U'$ 
%in $\mathcal T(F_v,\lambda)$
%by placing
%$\rho_{i_1}, \dotsc, \rho_{i_{\lambda_k}}$ in row $k$ of $U'$ so that
%$L(U') = R(U') =?$.  This is clearly a bijection.
%($\star$ State the bijection and justify its validity.)
%Thus the interpretation holds for $312$-avoiding permutations.
%By (\ref{eq:holdsforpavoidingtoo}) 
%it holds for \pavoiding permutations as well.
%($\star$ Prove the above here?, or prove a $q$-analog later, and just 
%refer to that result here?)

%\noindent
(iii) 
Applying (i) and Theorem~\ref{t:dsnuio} 
%and (i)
%($\star$ Explain how to combine
to Gasharov's~\cite[Thm.\,2]{GashInc},
we obtain the claimed interpretation for $312$-avoiding permutations.
% $w$ avoiding the pattern $312$.
(See also Section~\ref{s:chromsf}.)
% showed that for a unit interval
%order $P$, the numbers of column-strict and standard $P$-tableaux 
%of various shapes are related by the Kostka matrix. 
%(\ref{eq:thetaCimmB}),
%Theorem~\ref{t:dsnuio},
%and Stanley's definition of chromatic symmetric function?
Since standardness of $F_w$-tableaux depends only upon
$P(F_w)$, we may apply Theorem~\ref{t:thetavw}
%the argument preceeding (\ref{eq:holdsforpavoidingtoo})
to extend the result to \pavoiding permutations as well. 

%\noindent 
(iv-a) 
%To obtain the first interpretation,
Apply the definition (\ref{eq:pathmatrix}) of path matrix to 
the identity (\ref{eq:pimm}).

(iv-b) 
%To obtain the second interpretation,
%($\star$ Prove here.)
Let $v \sim w$ avoid the pattern $312$.
We define a map from
cylindrical tableaux in $\mathcal T(F_v,\lambda)$ 
%which are
%row-closed and left row-strict to
to cyclically row-semistrict tableaux in $\mathcal T(F_v,\lambda)$ having
%which are of 
type $e$ as follows. 
% and to
%it suffices to
%and simply 
%state a bijection from 
%The bijection is simple.  
For $U \in \mathcal T(F_v, \lambda)$ 
cylindrical with rows $k$ of $L(U)$ and $R(U)$ containing indices
$i_1, \dotsc, i_{\lambda_k}$ and
$i_2, \dotsc, i_{\lambda_k}, i_1$ (respectively) from left to right,
and $\rho = (\rho_1,\dotsc, \rho_n)$ the unique path family of type $e$
covering $F_v$,
create a cyclically row-semistrict tableau $U' \in \mathcal T(F_v, \lambda)$ 
by inserting $\rho_{i_1}, \dotsc, \rho_{i_{\lambda_k}}$ into row $k$,
from right to left.  
This map is bijective since 
in the descending star network $F_v$, 
there exists a (unique) path 
from source $i_j$ to sink $i_m$ if and only if $i_j > i_m$
or $\rho_{i_j}$ and $\rho_{i_m}$ intersect.
Thus the claimed interpretation holds for $312$-avoiding permutations.
Since cyclical row-semistrictness in $F_w$-tableaux of type $e$
depends only upon $P(F_w)$,
%we apply
%the argument preceeding (\ref{eq:holdsforpavoidingtoo}), 
we may apply Theorem~\ref{t:thetavw}
%It suffices to restrict our attention
%By Theorem~\ref{t:thetavw}, it holds for 
to extend the result to \pavoiding permutations as well.

%($\star$ State Matt's $\lambda_1 \cdots \lambda_r$ result with $q=1$ here?)

%(iv-c), (iv-d) Shareshian and Wachs~\cite[Sec.\,4]{SWachsChromQ}
%have shown that for $312$-avoiding permutations,
%these formulas are equivalent to Stanley's \cite[Thm.\,2.6]{StanSymm}.
%Since the claimed properties of $F_w$-tableaux depend only upon
%$P(F_w)$, we may apply Theorem~\ref{t:thetavw}
%to extend the result to \pavoiding permutations as well. 
%Statement (iv-d) alternatively follows from (iv-b).  It is clear that
%cyclically row-semistrict $F_w$-tableaux of type $e$ and shape $\lambda$
%may be counted by considering the subset that are right-anchored 
%and multiplying by $\lambda_1 \cdots \lambda_r$.  
%These are precisely the right-anchored row-semistrict $F_w$-tableaux
%of type $e$ and shape $\lambda$.
(iv-c) Shareshian and Wachs~\cite[Sec.\,4]{SWachsChromQ}
have shown that for $312$-avoiding permutations,
this formula is equivalent to Stanley's \cite[Thm.\,2.6]{StanSymm}.
Since the claimed property of $F_w$-tableaux depends only upon
$P(F_w)$, we may apply Theorem~\ref{t:thetavw}
to extend the result to \pavoiding permutations as well. 
(See also, \cite[Lem.\,6]{AthanPSE}.)

(iv-d) The number of tableaux in (iv-b) is equal to the cardinality
of the subset that are right-anchored, times $\lambda_1 \cdots \lambda_r$.
This subset is precisely
% alternatively follows from (iv-b).  It is clear that
%cyclically row-semistrict $F_w$-tableaux of type $e$ and shape $\lambda$
%may be counted by considering the subset that are right-anchored 
%and multiplying by $\lambda_1 \cdots \lambda_r$.  
%These are precisely 
the right-anchored row-semistrict $F_w$-tableaux
of type $e$ and shape $\lambda$.
Alternatively, we may use the Shareshian-Wachs argument of (iv-c).

%\noindent 
(v-a) This was first stated in \cite{CSSkanPathTab}, 
%for $312$-avoiding permutations,
and will be proved in
%See 
%This is a special case of 
Theorem~\ref{t:qphipartial}.
%($\star$ Prove here? or prove a $q$-analog later, and just refer to that result here?)
A different interpretation 
%of this case 
was given in 
\cite[Thm.\,2.5.1]{WolfgangThesis}.
 
%\noindent 
(v-b) 
Apply Lindstr\"om's Lemma and 
%Theorem~\ref{t:thetavw}
(\ref{eq:thetaCimmB}) 
to Stembridge's
identity (\ref{eq:stemmonrect}).
%, proved in \cite[Thm.\,2.8]{StemConj}.
\end{proof}

Conspicuously absent from Theorem~\ref{t:sncfinterp} is an
%a general
interpretation of monomial class function evaluations 
of the form $\phi^\lambda(C'_w(1))$ which holds for all
%for general 
$\lambda \vdash n$.
%when $w$ \avoidsp.
% which holds for all $\lambda \vdash n$.
As we have mentioned in the first table of Section~\ref{s:intro},
these integers are conjectured to be nonnegative.
The problem of interpreting them has been posed 
from 
%several 
different points of view 
by Haiman, Stanley and Stembridge 
\cite[Conj.\,2.1]{HaimanHecke},
\cite[Conj.\,5.1]{StanSymm},
\cite[Conj.\,5.5]{StanStemIJT},
\cite[Conj.\,2.1]{StemConj}.
Any extension of the statements
in Theorem~\ref{t:sncfinterp} (v-a), (v-b) would be interesting.
\begin{prob}\label{p:monevalinterp}
For (special cases of) $w \in \sn$ and $\lambda \vdash n$, 
find a combinatorial proof that $\phi^\lambda(C'_w(1))$ is nonnegative.
\end{prob}


%Haiman~\cite{HaimanHecke} and Stembridge~\cite{StemImm} have shown
%that we have $\chi^\lambda(C'_w(1)) \geq 0$ for all $\lambda \vdash n$
%and all $w \in \sn$.
%However, there is no conjectured combinatorial interpretation 
%for $\chi^\lambda(C'_w(1))$ unless $w$ \avoidsp.
%% $312$.
%Haiman~\cite{HaimanHecke} and Stembridge~\cite{StemConj} have also conjectured 
%that we have $\phi^\lambda(C'_w(1)) \geq 0$ for all $\lambda \vdash n$
%and all $w \in \sn$.
%There is no general conjectured combinatorial interpretation 
%for $\phi^\lambda(C'_w(1))$, even in the case that $w$ avoids $312$,
%unless $\lambda$ has one of the special forms stated in 
%Theorem~\ref{t:sncfinterp} (v-a), (v-b).

\ssec{Inversions in path tableaux}
For $\lambda \vdash n$ and
%$\theta$ an $\sn$-class function and 
$w \in \sn$ \avoidingp,
% $312$,
parts (i) -- (iv-d) of
Theorem~\ref{t:sncfinterp}
%Theorem~\ref{t:sncfinterp}~(i) -- (iv-d) 
interpret
%$\theta(C'_w(1))$
$\epsilon^\lambda(C'_w(1))$,
$\eta^\lambda(C'_w(1))$,
$\chi^\lambda(C'_w(1))$, and
$\psi^\lambda(C'_w(1))$
as cardinalities of certain sets of $F_w$-tableaux.
%\ssec{Interpretations of evaluations of $\hnq$-traces}
Using these same sets of $F_w$-tableaux and 
%certain statistics on $P$-tableaux,
variations of the permutation statistic $\inv$,
% on $P$-tableaux,
%define a statistic
%mapping tableaux to nonnegative integers, and 
we show 
%here and 
in Sections~\ref{s:hnqinterph}, 
\ref{s:hnqinterpe}, 
\ref{s:hnqinterps}, 
\ref{s:hnqinterpp}
%(or conjecture)
that 
$\epsilon_q^\lambda(\qew C'_w(q))$, 
$\eta_q^\lambda(\qew C'_w(q))$,
$\chi_q^\lambda(\qew C'_w(q))$, and
$\psi_q^\lambda(\qew C'_w(q))$
are generating functions for
tableaux on which the statistics take the values $k = 0,1,\dotsc$.
%In each case, our statistic is based upon the number of inversions
%of a permutation in $\sn$.
%For none of the traces $\theta_q$ did we have a known combinatorial
%interpretation of $\theta_q(\qew C'_w(q))$, until now.
%for $F$ a bijective skeleton.
%, as follows.
%have the following interpretations of class functions

Specifically, we adapt the permutation statistic 
$\inv$ for use on path tableaux as follows.
%our statistics will be variations on the well-known
%inversion statistic $\inv$ on permutations.
Let $\pi = (\pi_1, \dotsc, \pi_n)$ be a path family of type $v$ in some
zig-zag network $F$,
%$F$ be a descending star network, 
and let $U$ be a $\pi$-tableau.
%$U$ 
%containing path family 
%of type $v$.
%(Thus $\pi_i$ begins at source vertex $i$ and terminates at sink vertex $v_i$
%for $i = 1,\dotsc, n$.)
% of type $e$, call
Let $(\pi_i, \pi_j)$ be a pair of intersecting paths in $F$ such that 
$\pi_i$ appears in a column of $U$ to the left of the column containing $\pi_j$.
%an {\em inversion} in $U$ if 
%Call a pair $(\pi_i, \pi_j)$ an {\em inversion} in $U$ if 
Call $(\pi_i, \pi_j)$ a {\em (left) inversion} in $U$ if 
we have $i > j$
%$\pi_i$ intersects $\pi_j$ and we have
%\begin{enumerate}
%\item $\pi_i$ intersects $\pi_j$, 
%\item $i < j$, 
%\item $\pi_j$ appears in a column to the left of the column containing $\pi_j$.
%\end{enumerate}
%Call a pair $(\pi_i,\pi_j)$ a {\em right inversion} in $U$ if
and a {\em right inversion} in $U$ if
we have $v_i > v_j$.
%\begin{enumerate}
%\item $i < j$ and $w_i > w_j$,
%\item $\pi_i$ intersects $\pi_j$,
%\item $\pi_i$ appears in a column to the left of the column containing $\pi_j$.
%\end{enumerate}
Let $\inv(U)$ denote the number of 
%left 
inversions in $U$,
and let $\rinv(U)$ denote the number of right inversions in $U$.
%Clearly if $\type(\pi) = e$ then we have $\inv(U) = \rinv(U)$ in 
%any $\pi$-tableau $U$.
%\begin{thm}
%Let $w \in \sn$ avoid the 
%pattern $312$.  
%%patterns $3412$, $4231$.  
%For $\lambda \vdash n$
%we have
%\begin{equation}\label{eq:chiqinterp}
%\chi_q^\lambda(\qew C'_w(q)) = \sum_U q^{\inv(U)},
%\end{equation}
%where the sum is over all standard $F_w$-tableaux of type $e$ and 
%shape $\lambda^\tr$.
%\end{thm}
%\begin{proof}
%Let $(a_\lambda(q))_{\lambda \vdash n}$
%and $(b_\lambda(q))_{\lambda \vdash n}$ 
%be the sequences of polynomials defined by the 
%right-hand sides of Equations (\ref{eq:epqinterp}) and (\ref{eq:chiqinterp}).
%%right-hand sides 
%Following Gasharov's proof~\cite{GashInc} 
%of the interpretation of $\chi^\lambda(C'_w(1))$ 
%in Proposition~\ref{p:sncfinterp},
%Shareshian and Wachs~\cite{SWachsChromQ}
%proved that these sequences are related by the transition matrix
%of Kostka numbers which also relates 
%the elementary symmetric functions and Schur functions indexed by partitions
%$\lambda$ of $n$.  This is precisely the transition matrix which relates the
%characters on the left-hand sides of the two equations.
%\end{proof}
%($\star$ Do an example.)
%Let $U$ be an $F$-tableau of shape $\lambda = (\lambda_1, \dotsc, \lambda_r)$
%containing a path family $\pi$, 
%and let 

Sometimes we will compute inversions in a one-rowed tableau formed by
concatenating all of the rows of a path tableau $U$.
Let $U_i$ be the $i$th row of $U$, and
let $U_1 \circ \cdots \circ U_r$ and $U_r \circ \cdots \circ U_1$
be the $F$-tableaux of shape $n$ consisting of the rows of $U$ concatenated
in increasing and decreasing order, respectively.
% and let 
%$U_{i,j}$ 
%$\pi_{i,1}, \dotsc, \pi_{i,\lambda_i}$ 
%denote the path in row $i$, position $j$ of $U$. 
%Define the {\em flattened $F$-tableau} $\ol{U}$ corresponding to $U$ to be 
%the tableau of shape $n$ which is the concatenation of rows $1,\dotsc,r$,
%\begin{equation}\label{eq:flattab}
%%\ol U = (\pi_{1,1}, \dotsc, \pi_{1,\lambda_1}, \pi_{2,1}, \dotsc, \pi_{r,\lambda_r}).
%\ol U = (U_{1,1}, \dotsc, U_{1,\lambda_1}, 
%U_{2,1}, \dotsc, U_{2,\lambda_2}, 
%\dotsc,
%U_{r,1}, \dotsc, U_{r,\lambda_r}).
%\end{equation}
We will also compute inversions in the {\em transpose} $U^\tr$ of a 
path tableau $U$, whose rows are the columns of $U$.
It is easy to see that inversions in these one-rowed and transposed tableaux
are related by the identities
%equations
%any $\pi$-tableau $U$ of shape 
%$\lambda = (\lambda_1, \dotsc, \lambda_r)$ 
%satisfies the identities
\begin{equation}\label{eq:invdecomp}
\begin{gathered}
\inv(U_1 \circ \cdots \circ U_r) 
= \inv(U_1) + \cdots + \inv(U_r) + \inv(U^\tr),\\
\rinv(U_1 \circ \cdots \circ U_r) 
= \rinv(U_1) + \cdots + \rinv(U_r) + \rinv(U^\tr).
\end{gathered}
\end{equation}

\section{
%Statistics on $F$-tableaux and %combinatorial 
Interpretation of $\eta_q^\lambda(\qew C'_w(q))$}
%of $\hnq$-traces}
\label{s:hnqinterph}


%Let $U$ be an $F$-tableau of shape $\lambda = (\lambda_1, \dotsc, \lambda_r)$
%containing a path family $\pi$, 
%and let $U_i$ be the $i$th row of $U$.
%Let $U_1 \circ \cdots \circ U_r$ and $U_r \circ \cdots \circ U_1$
%be the $F$-tableaux of shape $n$ consisting of the rows of $U$ concatenated
%in increasing and decreasing order, respectively.
% and let 
%$U_{i,j}$ 
%$\pi_{i,1}, \dotsc, \pi_{i,\lambda_i}$ 
%denote the path in row $i$, position $j$ of $U$. 
%Define the {\em flattened $F$-tableau} $\ol{U}$ corresponding to $U$ to be 
%the tableau of shape $n$ which is the concatenation of rows $1,\dotsc,r$,
%\begin{equation}\label{eq:flattab}
%%\ol U = (\pi_{1,1}, \dotsc, \pi_{1,\lambda_1}, \pi_{2,1}, \dotsc, \pi_{r,\lambda_r}).
%\ol U = (U_{1,1}, \dotsc, U_{1,\lambda_1}, 
%U_{2,1}, \dotsc, U_{2,\lambda_2}, 
%\dotsc,
%U_{r,1}, \dotsc, U_{r,\lambda_r}).
%\end{equation}

%\subsection{Interpretation of $\eta_q^\lambda(\qew C'_w(q))$} 
%\ssec{Interpretation of $\eta_q^\lambda(\qew C'_w(q))$} 
%Let us first consider a combinatorial interpretation for
%$\eta_q^\lambda(\qew C'_w(q))$ 
%Now consider proving Conjecture~\ref{conj:qh}.
%the first conjecture.
%where 
Let $w \in \sn$ \avoidp, and
% the pattern $312$, and
%-avoiding permutation.
% corresponding
%to unit interval order $P$ and descending star network $\hat F$.
let $B$ be the path matrix of $F_w$.
%antiadjacency matrix of $P$ corresponding to an ar-labeling
%of $P$.  
Using (\ref{eq:qinducedimmssubmat}) and Proposition~\ref{p:qimmchar}, 
we have
%observe that the 
%left-hand side of (\ref{eq:conjeqlamb}) is
%Now let us compute $\epsilon_q^\lambda(\qew \beta_q(\hat F))$.
%This is 
\begin{equation}\label{eq:hqlambeta}
%\begin{aligned}
\eta_q^\lambda(\qew C'_w(q)) 
= \sigma_B(\imm{\eta_q^\lambda}(x)) 
%\\
%\sum_{v\leq w} \qew \epsilon_q^\lambda (\wT_v) \qivw
%= \sum_{v \leq w} \epsilon_q^\lambda (\wT_v) \qev.
%\end{equation}
%%where the sum is over only $v$ satisfying $\a_{1,v_1} \cdots a_{n,v_n} = 1$.
%Since we have
%\begin{equation*}
%\imm{\epsilon_q^\lambda}(x) 
%= \sum_{v \in \sn} \epsilon_q^\lambda (\wT_v) \permmon xv
%&= \sigma_B \Big (\sum_{(I_1, \dotsc, I_r)} \qper(x_{I_1,I_1}) \cdots \qper(x_{I_r,I_r}) \Big)\\
= \sum_{(I_1, \dotsc, I_r)} \sigma_B (\qper(x_{I_1,I_1}) \cdots \qper(x_{I_r,I_r}) ),
%\end{aligned}
\end{equation}
where the sum is over all ordered set partitions $(I_1, \dotsc, I_r)$ of
$[n]$ of type $\lambda = (\lambda_1, \dotsc, \lambda_r)$.
%which satisfy $|I_j| = \lambda_j$ for $j = 1,\dotsc,r$.
%\begin{question}
%What are the two quantum permanent versions of 
%Conjecture~\ref{c:esingleterm}
%and the two quantum permanent versions of Conjecture~\ref{c:nonprincprod}?
%\end{question}
%By Equation~(\ref{eq:altqimmeh}), the last expression in
%(\ref{eq:hqlambeta}) can be rewritten as
Let $\slambda$ denote the Young subgroup of $\sn$ generated by
\begin{equation*}
%\label{eq:youngsubgen}
%J \defeq 
\{ s_1, \dotsc, s_{n-1} \} \ssm 
\{ s_{\lambda_1}, s_{\lambda_1 + \lambda_2}, 
s_{\lambda_1 + \lambda_2 + \lambda_3}, 
\dotsc, 
s_{n-\lambda_r} \},
\end{equation*}
and let $\slambdamin$ be the set of Bruhat-minimal representatives of cosets
of the form $\slambda u$, i.e., the elements $u \in \sn$ for which each
of the subwords
\begin{equation}\label{eq:subwordsofu}
u_1 \cdots u_{\lambda_1}, \qquad 
u_{\lambda_1 +1} \cdots u_{\lambda_1 + \lambda_2}, \quad
\dotsc, \quad
u_{n-\lambda_r + 1} \cdots u_n
\end{equation}
is strictly increasing.
%(See, e.g., \cite{CurtisLIT}.)
%satisfying
%\begin{equation*}
%u_1 < \cdots < u_\lambda, \qquad u_{\lambda_1 +1} < \cdots < u_{\lambda_1 + \la%mbda+2}, 
%\dotsc, u_{n-\lambda_r + 1} < \cdots < u_n.
%\end{equation*}
It is clear that such elements correspond bijectively to the 
ordered set partitions $(I_1, \dotsc, I_r)$ in (\ref{eq:hqlambeta}).
Expanding the product of permanents, we obtain monomials of the form
$\quv$ times
%$q_{u,v} x^{u,v}$ 
\begin{equation*}
x^{u,v} \defeq \doublepermmon xuv,
\end{equation*}
%$q_{u,v}\doublepermmon xuv$
where $v$ is the concatenation, in order, of 
rearrangements of the $r$ words (\ref{eq:subwordsofu}).  
Thus $v$ may be written as $yu$ with $y \in \slambda$,
or as $uy$ with $y \in u^{-1}\slambda u$.
%To economize notation, we will abbreviate
%\begin{equation*}
%\end{equation*}
%Now we may rewrite the
%We may therefore rewrite the 
%last 
Now the sum in 
%(\ref{eq:hqlambeta}) as
(\ref{eq:hqlambeta}) becomes
\begin{equation}\label{eq:hqlambetaalt}
%\begin{gathered}
%\imm{\epsilon^\lambda}(x) 
%= \sum_{u \in \Wlem} \sum_{y \in S_{I(u)}} (-1)^{\ell(y)} \qiey x^{u,uy},\\
%= \sum_{u \in \Wlem} \sum_{y \in S_{I(u)}} (-1)^{\ell(y)} \qiey x^{e,y},\\
%\imm{\eta^\lambda}(x) 
%= 
%\sum_{u \in \Wlem} \sum_{y \in S_{I(u)}} \sigma_B (q_{u,uy} x^{u,uy}).
\sum_{u \in \slambdamin} \sum_{y \in u^{-1}\slambda u} \sigma_B (q_{u,uy} x^{u,uy}).
%= \sum_{u \in \Wlem} \sum_{y \in S_{I(u)}} \qey x^{e,y},
%\end{gathered}
\end{equation}
Let us therefore consider evaluations of the form 
%\begin{equation*}
%\label{eq:sigmabquv}
$\sigma_B (\quv x^{u,v})$.
%\end{equation*}

To combinatorially interpret these evaluations, 
%we consider 
%path families and certain orderings of these in tableaux.
let $\pi = (\pi_1, \dotsc, \pi_n)$ be a path family 
(of arbitrary type)
which covers a zig-zag
%descending star 
network $F$, and
define $U(u,\pi)$ to be the $\pi$-tableau of shape $(n)$
containing $\pi$ in the order $\pi_{u_1}, \dotsc, \pi_{u_n}$.
Clearly the left tableau of $U(u,\pi)$ is $u_1 \cdots u_n$.
If the right tableau is $v_1 \cdots v_n$ then $\pi$ has type $u^{-1}v$.
If $s_i$ is a left descent for $u$, then right inversions in $U(u,\pi)$
and $U(s_iu,\pi)$ are related as follows.

\begin{prop}\label{p:usupi}
Fix $u, v \in \sn$, 
%with $s_iu < u$, 
let $F$ be a zig-zag
%descending star 
network, and let $\pi = (\pi_1, \dotsc, \pi_n)$ be a path family
of type $u^{-1}v$ which covers $F$. 
If $s_i u < u$ then we have
\begin{equation*}
\rinv(U(u,\pi)) = 
\begin{cases}
\rinv(U(s_iu,\pi)) - 1 &\text{if $s_i v > v$,} \\
\rinv(U(s_iu,\pi)) 
&\parbox[t]{.5 \textwidth}
{if $s_iv < v$ and no path family
of type $u^{-1}s_iv$ covers $F$,}\\
\rinv(U(s_iu,\pi)) + 1 
&\parbox[t]{.5 \textwidth}
{if $s_iv < v$ and some path family
of type $u^{-1}s_iv$ covers $F$.}
\end{cases}
\end{equation*}
\end{prop}
\begin{proof}
The tableaux $U(u,\pi)$ and $U(s_iu, \pi)$ are identical except 
that $\pi_{u_i}$ appears before $\pi_{u_{i+1}}$ in $U(u,\pi)$.
%for the paths in positions $i$ and $i+1$.
Thus we have
\begin{equation*}
\rinv(U(u,\pi)) = \begin{cases}
\rinv(U(s_iu,\pi)) - 1 
&\parbox[t]{.5 \textwidth}
{if $(\pi_{u_i}, \pi_{u_{i+1}})$ is a right inversion 
in $U(s_iu,\pi)$ but not in $U(u,\pi)$,}\\
\rinv(U(s_iu,\pi)) + 1 
&\parbox[t]{.5 \textwidth}
{if $(\pi_{u_i}, \pi_{u_{i+1}})$ is a right inversion 
in $U(u,\pi)$ but not in $U(s_iu,\pi)$,}\\
\rinv(U(s_iu,\pi)) 
&\parbox[t]{.5 \textwidth}
{if $(\pi_{u_i}, \pi_{u_{i+1}})$ is not a right inversion 
in $U(s_iu,\pi)$ or $U(u,\pi)$.}
\end{cases}
\end{equation*}
Since $s_i u < u$, we have $u_i > u_{i+1}$.  

If $s_iv > v$, then we have $v_i < v_{i+1}$, and 
Observation~\ref{o:pnetintersect} implies that the paths
$\pi_{u_i}$ and $\pi_{u_{i+1}}$ intersect, forming
%Since $\pi_{u_{i+1}}$ appears
%first in $U(s_iu,\pi)$, 
%Thus these paths form 
a right inversion in $U(s_iu,\pi)$ and not in $U(u,\pi)$.

If $s_iv < v$, then we have $v_i > v_{i+1}$, and the paths
$\pi_{u_i}$ and $\pi_{u_{i+1}}$ 
do not form a right inversion in $U(s_iu,\pi)$.
Suppose that some path family $\pi' = (\pi'_1, \dotsc, \pi'_n)$
of type $u^{-1}s_iv$ covers $F$.  
%Then the tableau of type $u^{-1}s_iv$ which covers $F$.  
Then the tableau
$U(u,\pi')$ satisfies 
\begin{equation*}
L(U(u,\pi')) = (u_1, \dotsc, u_n), 
\qquad
R(U(u,\pi')) = (v_1, \dotsc, v_{i-1}, v_{i+1}, v_i, v_{i+2}, \dotsc, v_n ).
\end{equation*}
By the uniqueness of source-to-sink paths in zig-zag
%descending star 
networks,
this tableau is identical to the tableau $U(u,\pi)$ except for the paths
$\pi'_{u_i}$ and $\pi'_{u_{i+1}}$ in positions $i$ and $i+1$, which terminate
at sinks $v_{i+1} < v_i$, respectively.
By Observation~\ref{o:pnetintersect}, the paths $\pi'_{u_i}, \pi'_{u_{i+1}}$
cross and the paths $\pi_{u_i}$, $\pi_{u_{i+1}}$ intersect,
forming a right inversion in the tableau $U(u,\pi)$.
On the other hand, suppose that no path family of type $u^{-1}s_iv$ 
covers $F$.
Since $\pi$ has type $u^{-1}v$,  
%Again by the uniqueness of source-to-sink paths in zig-zag
%descending star 
%networks,
we can deduce that either there is no path in $F$ 
from source $u_i$ to sink $v_{i+1}$
or there is no path from source $u_{i+1}$ to sink $v_i$.
By Observation~\ref{o:pnetintersect}, the paths $\pi_{u_i}$, $\pi_{u_{i+1}}$ 
do not intersect and therefore do not form a right inversion in $U(u,\pi)$. 
\end{proof}
 
Now we evaluate $\sigma_B(\quv x^{u,v})$, first in the case that $u = e$.
\begin{prop}\label{p:sigaqevxev}
Let $w$ in $\sn$ \avoidp,
% the pattern $312$,
%$\hat F$ be the descending star network 
%corresponding to the $312$-avoiding permutation 
%let $A$ be the antiadjacency matrix 
%of the unit interval order corresponding to $\hat F$,
%Let $\hat F$ be the descending star network corresponding to the $312$-avoiding
%permutation $w$ in $\sn$, 
let $B$ be the path matrix of $F_w$,
and fix $v$ in $\sn$.  Then we have
\begin{equation*}
\sigma_B (\qev x^{e,v}) = 
\begin{cases}
q^{\rinv(U(e,\pi))} &\text{if there exists a path family $\pi$ of type $v$ which covers $F_w$},\\
0 &\text{otherwise}.
\end{cases}
\end{equation*}
\end{prop}
\begin{proof}
By definition we have
\begin{equation}\label{eq:qlvaev}
%\begin{aligned}
\sigma_B (\qev x^{e,v}) = \qev \qev b^{e,v} = q^{\ell(v)}b^{e,v}.
\end{equation}
First assume that there exists a (unique) path family $\pi$
% = (\pi_1, \dotsc, \pi_n)$ 
of type $v$ that covers $F_w$.
Then we have $b^{e,v} = 1$.  In the tableau $U(e,\pi)$, 
paths appear in the order $(\pi_1, \dotsc, \pi_n)$.
%of this family appear in order in 
Now observe that for each inversion in $v$, i.e., 
each pair $(v_i, v_j)$ with $i < j$ and $v_i > v_j$,
the paths $\pi_i$ (from source $i$ to sink $v_i$)
and $\pi_j$ (from source $j$ to sink $v_j$) cross in $F_w$
and therefore form a right inversion in $U(e,\pi)$. 
Conversely, for each noninversion $(v_i, v_j)$ in $v$, the paths
$\pi_i$ and $\pi_j$ do not form a right inversion in $U(e,\pi)$.
Thus we have $\ell(v) = \rinv(U(e,\pi))$, and
the expression in (\ref{eq:qlvaev}) is equal to 
%we have $q^{\ell(v)}a^{e,v} = 
$q^{\rinv(U(e,\pi))}$.

Now assume that there is no path family of type $v$ which covers $F_w$.
Then we have $b^{e,v} = 0$ and
the expressions in (\ref{eq:qlvaev}) are equal to $0$.
\end{proof}

More generally, we evaluate $\sigma_B(\quv x^{u,v})$ as follows.
%have the following. 
%($\star$ Check the following
%results to ensure that they hold for \pavoiding permutations.)
\begin{prop}\label{p:sigaquvxuv}
Let $w$ in $\sn$ \avoidp,
% the pattern $312$,
let $B$ be the path matrix of $F_w$,
%Let $\hat F$ be the descending star network 
%corresponding to the $312$-avoiding permutation $w$ in $\sn$, 
%let $A$ be the antiadjacency matrix 
%of the unit interval order corresponding to $\hat F$,
and fix $u$, $v$ in $\sn$.  Then we have
\begin{equation*}
%\label{eq:sigaquvxuv}
\sigma_B (\quv x^{u,v}) = 
\begin{cases}
q^{\rinv(U(u,\pi))} &\text{if there exists a path family $\pi$ of type $u^{-1}v$ 
covering $F_w$},\\
0 &\text{otherwise}.
\end{cases}
\end{equation*}
\end{prop}
\begin{proof}
We use induction on the length of $u$.
By Proposition~\ref{p:sigaqevxev}, the claimed formula 
%(\ref{eq:sigaquvxuv}) 
holds when $u$ has length $0$.
Now assume that the formula holds when $u$ has length $1, \dotsc, k-1$,
and consider $u$ of length $k$.
Choosing a left descent $s_i$ of $u$, we may write
\begin{equation*}
\begin{aligned}
\sigma_B (\quv x^{u,v}) &= 
\begin{cases}
\sigma_B (\quv x^{s_iu, s_iv}) &\text{if $s_i v > v$,} \\
\sigma_B (\quv x^{s_iu, s_iv} + (\qdiff) \quv x^{s_iu,v} ) &\text{if $s_i v < v$,}
\end{cases}\\
&= \begin{cases}
q^{-1} \sigma_B (q_{s_i u, s_i v} x^{s_iu, s_iv}) &\text{if $s_i v > v$,} \\
%\sigma_B (q_{s_i u, s_i v} x^{s_iu, s_iv} + (1 - q^{-1}) q_{s_iu, v} x^{s_iu,v}) 
\sigma_B (q_{s_i u, s_i v} x^{s_iu, s_iv}) + (1 - q^{-1}) \sigma_B (q_{s_iu, v} x^{s_iu,v}) 
&\text{if $s_i v < v$.}
\end{cases}
\end{aligned}
\end{equation*}

Suppose first that we have $s_i v > v$.
Then 
%$(s_i u)^{-1} (s_i v) = u^{-1}v$ and 
by induction we have 
\begin{equation*}
%q^{-1} 
\sigma_B (q_{s_i u, s_i v} x^{s_iu, s_iv}) = 
%\sigma_B (\quv x^{u,v}) = 
\begin{cases}
%q^{\rinv(U(s_i u,\pi))-1} 
q^{\rinv(U(s_i u,\pi))} 
&\parbox[t]{.5 \textwidth}{if there exists a path family $\pi$ of 
type $u^{-1}v$ covering $F_w$,}\\
0 &\text{otherwise,}
\end{cases}
\end{equation*}
and Proposition~\ref{p:usupi} implies that the claim is true in this case.

Now suppose that we have $s_i v < v$
and consider path families of types $u^{-1}v$ and $u^{-1}s_iv$ 
which cover $F_w$.
If there are no path families of types $u^{-1}v$ and $u^{-1}s_iv$
which cover $F_w$, then
by induction we have 
\begin{equation}\label{eq:sigaaeqx1}
%\sigma_B (q_{s_i u, s_i v} x^{s_iu, s_iv} + (1 - q^{-1}) q_{s_iu, v} x^{s_iu,v}) = 0.
\sigma_B (q_{s_i u, s_i v} x^{s_iu, s_iv}) = \sigma_B (q_{s_iu, v} x^{s_iu,v}) = 0.
\end{equation}
If there exists a path family $\pi = (\pi_1,\dotsc,\pi_n)$ of type
$u^{-1}v$ which covers $F_w$, but no path family of type $u^{-1}s_iv$
which covers $F_w$, then by induction and Proposition~\ref{p:usupi}
we have
\begin{equation}\label{eq:sigaaeqx2}
\begin{gathered}
\sigma_B (q_{s_i u, s_i v} x^{s_iu, s_iv}) = q^{\rinv(U(s_iu,\pi))} = q^{\rinv(U(u,\pi))},\\
\sigma_B (q_{s_iu, v} x^{s_iu,v}) = 0.
\end{gathered}
\end{equation}
If there exists a path family
$\pi' = (\pi'_1,\dotsc, \pi'_n)$ 
of type $u^{-1}s_iv$ which covers $F_w$,
%Then the pair $(\pi_{u_i}, \pi_{u_{i+1}})$ is a right inversion in  
%then by the argument in the proof of Proposition~\ref{p:usupi},
then the paths $\pi'_{u_i}$, $\pi'_{u_{i+1}}$ 
from sources $u_i > u_{i+1}$ to sinks $v_{i+1} < v_i$ (respectively)
%and therefore cross.
cross.
It follows that there exists a path family 
$\pi = (\pi_1, \dotsc, \pi_n)$ of type $u^{-1}v$
which covers $F_w$, and which agrees with $\pi'$ except that
%and the pair $(\pi_{u_i}, \pi_{u_{i+1}})$ is an
%inversion in the tableau $U(u,\pi)$.
%In particular, the path families are identical except that
the paths $\pi_{u_i}$, $\pi_{u_{i+1}}$ from sources $u_i > u_{i+1}$
to sinks $v_i > v_{i+1}$ (respectively)
intersect but do not cross.
%while
%$\pi'_{u_i}$, $\pi'_{u_{i+1}}$ terminate at sinks $v_{i+1}$, $v_i$, respectively.
%Thus 
%$(s_i u)^{-1} (s_i v) = u^{-1}v$ and 
By induction and the existence of $\pi'$ and $\pi$ we have
%, and by induction, we have 
\begin{equation*}
%\label{eq:sigaaeqx3}
\begin{gathered}
\sigma_B (q_{s_i u, s_i v} x^{s_iu, s_iv}) = q^{\rinv(U(s_i u,\pi))},\\ 
%+ (1 - q^{-1}) \sigma_B (q_{s_iu, v} x^{s_iu,v}) = 
%+ (1 - q^{-1}) q_{s_iu, v} x^{s_iu,v}) = 
\sigma_B (q_{s_iu, v} x^{s_iu,v}) = 
%q^{\rinv(U(s_i u,\pi))} + q^{\rinv(U(s_iu, \pi'))} - q^{\rinv(U(s_iu, \pi'))-1}. 
q^{\rinv(U(s_iu, \pi'))}. 
\end{gathered}
\end{equation*}
%Now consider the tableaux
%$U(u,\pi)$, 
%$U(s_i u, \pi)$ and $U(s_i u, \pi')$, which agree except for the
%pairs of (intersecting) paths in positions $i$ and $i+1$:
%paths $\pi_{u_{i+1}}, \pi_{u_i}$ terminate at sinks $v_{i+1}, v_i$,  
%respectively,
%while paths $\pi'_{u_{i+1}}, \pi'_{u_i}$ terminate at sinks $v_i, v_{i+1}$,
%respectively.
%Since $v_i > v_{i+1}$ we have
The tableaux 
$U(s_i u, \pi)$ and $U(s_i u, \pi')$ agree except in positions $i$ and $i+1$,
where paths $\pi'_{u_{i+1}}$ and $\pi'_{u_i}$ form a right inversion,
but $\pi_{u_{i+1}}$ and $\pi_{u_i}$ do not.
This fact and Proposition~\ref{p:usupi} imply that we have
\begin{equation*}
%\label{eq:uupi}
%\rinv(U(u,\pi)) = 
\rinv(U(s_iu, \pi')) = 
\rinv(U(s_iu, \pi)) + 1 = 
\rinv(U(u,\pi)),
\end{equation*}
%which by Proposition~\ref{p:usupi} is equal to $\rinv(U(u,\pi))$.
%Equations (\ref{eq:sigaaeqx3}) and 
%(\ref{eq:uupi}) 
%implies
%Thus we have
and
\begin{equation}\label{eq:sigaaeqx4}
\begin{gathered}
\sigma_B (q_{s_i u, s_i v} x^{s_iu, s_iv}) = q^{\rinv(U(u,\pi))-1},\\ 
(1 - q^{-1})\sigma_B (q_{s_iu, v} x^{s_iu,v}) = 
%(1 - q^{-1}) q^{\rinv(U(u, \pi))} =
q^{\rinv(U(u, \pi))} - q^{\rinv(U(u, \pi))-1}. 
\end{gathered}
%q^{\rinv(U(s_i u,\pi))} + q^{\rinv(U(s_iu, \pi'))} - q^{\rinv(U(s_iu, \pi'))-1}
%= q^{\rinv(U(u,\pi))},
\end{equation} 
Thus by Equations~(\ref{eq:sigaaeqx1}), 
(\ref{eq:sigaaeqx2}), and (\ref{eq:sigaaeqx4})
the claim is true when $s_iv < v$.
\end{proof}

Now we have the following $q$-analog of Theorem~\ref{t:sncfinterp} (ii-a).

\begin{thm}\label{t:qeta}
Let $w \in \sn$ \avoidp.
% the pattern $312$.
% corresponding
%to unit interval order $P$ and descending star network $\hat F$.
Then for $\lambda = (\lambda_1, \dotsc, \lambda_r) \vdash n$ we have
\begin{equation*}
\eta_q^\lambda(\qew C'_w(q)) = \sum_U q^{\rinv(U_1 \circ \cdots \circ U_r)},
\end{equation*}
where the sum is over all row-closed, left row-strict 
$F_w$-tableaux 
%$U$ 
of shape $\lambda$.
\end{thm}
\begin{proof}
Let $B$ be the path matrix of $F_w$ and let $(I_1,\dotsc,I_r)$ be a set
partition of $[n]$ of type $\lambda$.
%satisfying $|I_j| = \lambda_j$ for $j = 1,\dotsc,r$.
By 
%the comment following 
(\ref{eq:subwordsofu}) -- (\ref{eq:hqlambetaalt}),
%Proposition~\ref{p:qimmchar} and 
%(\ref{eq:hqlambetaalt}),
there is a permutation $u \in \slambdamin$ 
corresponding to $(I_1,\dotsc,I_r)$ such that 
we have
\begin{equation*}
%\eta_q^\lambda(\qew C'_w(q)) = \sum_{u \in \slambdamin} 
\sigma_B(\qper(x_{I_1,I_1}) \cdots \qper(x_{I_r,I_r})) = 
\sum_{y \in u^{-1}\slambda u} \sigma_B ( q_{u,uy} x^{u, uy} ).
\end{equation*}
%Now consider one permutation $u \in \slambdamin$ 
%and define the ordered set partition 
%\begin{equation*}
%I = I(u) = (I_1, \dotsc, I_r)
%\end{equation*}
%as in (\ref{eq:Iudef}).
By Proposition~\ref{p:sigaquvxuv}, this is equal to
%e inner sum may be rewritten as
%we have
\begin{equation}\label{eq:ypirinvuupi}
%\sum_{y \in u^{-1}\slambda u} \sigma_B ( q_{u,uy} x^{u, uy} ) = 
%\sum_{u \in \slambdamin} 
\sum_{(y,\pi)} q^{\rinv(U(u,\pi))},
\end{equation}
where the 
%last 
%inner sum now 
sum is over pairs $(y,\pi)$ such that 
%$y \in S_I$ 
$y \in u^{-1}\slambda u$ and $\pi$ is a path family of type $y$ 
which covers $F$.
If such a path family $\pi$ exists for a given permutation $y$, 
it is necessarily unique.  
Thus as $y$ varies over $u^{-1}\slambda u$
%As $u$ varies over $\slambdamin$ and
%.  Letting 
%$(y,\pi) = (y, \pi(y))$ varies as above, 
we have that
%the tableau 
$U(u, \pi)$
% = U(u,\pi(y))$ 
varies over all bijective path tableaux
of shape $(n)$ which satisfy
\begin{enumerate}
\item For $j=1,\dotsc,r$, 
%the sequence of indices of 
%the indices of 
the paths in positions 
$\lambda_1 + \cdots + \lambda_{j-1}+1, \dotsc,$ 
$\lambda_1 + \cdots + \lambda_j$ are indexed by $I_j$, in increasing order.
%the increasing rearrangement if $I_j$.
%increase: 
%$u_{\lambda_1 + \cdots + \lambda_{j-1}+1} < \cdots < u_{\lambda_1 + \cdots + \lambda_j}$.
\item The sequence of sink indices of these same paths 
are a rearrangement of $I_j$.
%permutation of the indices of the paths.
\end{enumerate}
Thus the expression
%over all $u \in \Wlem$ of the second sum 
in (\ref {eq:ypirinvuupi})
%(\ref{eq:ypirinvuupi})
%By Lemma ?, this 
may be rewritten as
\begin{equation}\label{eq:rinvudotsu}
\sum_U q^{\rinv(U_1 \circ \cdots \circ U_r)},
\end{equation}
where this last sum is over all 
row-closed, left row-strict 
$F_w$-tableaux $U$ of shape $\lambda$
for which path indices of $U_j$ are $I_j$ for $j=1,\dotsc,r$.
Summing over ordered set partitions and using (\ref{eq:hqlambeta}),
we have the desired result.
%which satisfy $L(U) = (u_1, \dotsc, u_n)$.
\end{proof}

%($\star$ Do an example.)
For example, consider the network
\begin{equation}\label{eq:F3421}
F_{3421} = \raisebox{-10mm}{
\includegraphics[height=20mm]{xfigures/xstars1n}}\ .
\end{equation}
It is easy to verify that 
there are twenty row-closed, left row-strict
$F_{3421}$-tableaux of shape $31$.
Four of these are
\begin{equation*}
%\label{eq:etatabs31}
\raisebox{-6mm}{\includegraphics[height=12mm]{tableaux/31_44122133}}\ ,\quad
\raisebox{-6mm}{\includegraphics[height=12mm]{tableaux/31_44132132}}\ ,\quad
\raisebox{-6mm}{\includegraphics[height=12mm]{tableaux/31_33122441}}\ ,\quad
\raisebox{-6mm}{\includegraphics[height=12mm]{tableaux/31_22133441}}\ ,
\end{equation*}
where $i,j$ represents the unique path from source $i$ to sink $j$.
These tableaux $U$ of shape $31$
%and if we construct from each tableau $U$ 
yield tableaux $U_1 \circ U_2$ of shape $4$,
\begin{equation*}
%\label{eq:etatabs4}
\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_12213344}}\ ,\quad
\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_13213244}}\ ,\quad
\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_12244133}}\ ,\quad
\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_13344122}}\ ,
\end{equation*}
which have $1$, $2$, $3$, and $4$ right inversions, respectively.
Together, 
%they
the tableaux 
contribute $q + q^2 + q^3 + q^4$
to $\eta_q^{31}(q_{e,3421} C'_{3421}(q)) = 1 + 3q + 6q^2 + 6q^3 + 3q^4 + q^5$.
%\begin{cor}
%Let $w \in \sn$ \avoidp,
%let $B$ be the path matrix of $F_w$,
%and let $(I_1,\dotsc,I_r)$ be an ordered set partition of $[n]$.
%Then we have
%\begin{equation*}
%\sigma_B(\qper(x_{I_1,I_1}) \cdots \qper(x_{I_r,I_r})) = 
%\sum_U q^{\rinv(U_1 \circ \cdots \circ U_r)},
%\end{equation*}
%where the sum is over all row-closed, left row-strict 
%$F_w$-tableaux $U$ of shape $(|I_1|, \dotsc, |I_r|)$
%for which path indices in $U_j$ are $I_j$,
%for $j = 1,\dotsc,r$.
%\end{cor}
%($\star$ Incorporate this into proof of theorem?)

Expanding $\eta_q^\lambda$ in terms of irreducible characters
and Kostka numbers, $\eta_q^\lambda = \sum K_{\mu,\lambda} \chi_q^\mu$,
%where $\{ K_{\mu,\lambda} \,|\, \mu, \lambda \vdash n \}$ are the Kostka numbers,
and using Haiman's result~\cite[Lem\,1.1]{HaimanHecke},
we have that the coefficients of $\eta_q^\lambda( \qew C'_w(q))$ 
are symmetric and unimodal
about $\qew$ for all $w \in \sn$.
In the case that $w$ \avoidsp, it would be interesting to explain
this phenomenon combinatorially in terms of Theorem~\ref{t:qeta}.
%($\star$ Mention that it

%The authors know of no extension of
It would also be interesting to extend 
Theorem~\ref{t:qeta} to include 
a $q$-analog of Theorem~\ref{t:sncfinterp} (ii-b).
%It would be interesting to find such an extension 
In particular, the identity
\begin{equation*}
\eta_q^{(n)} = \sum_{\lambda \vdash n} \phi_q^\lambda
\end{equation*}
suggests that an answer to Problem~\ref{p:monevalinterp} and its $q$-analog
are related to a set partition of tableaux counted by $\eta^{(n)}$.
It is not clear whether such a partition is more easily expressed
in terms of left row-strict tableaux of shape $(n)$, 
or row-semistrict tableaux of type $e$ and shape $(n)$.
%)
\begin{prob}
Find a statistic $\STAT$ on $F$-tableaux such that we have
\begin{equation*}
\eta_q^\lambda(\qew C'_w(q)) = \sum_U q^{\STAT(U)},
\end{equation*}
where the sum is over all row-semistrict $F_w$-tableaux 
of type $e$ and shape $\lambda$.
\end{prob}

%($\star$ Introduce this.)
As a consequence of Theorem~\ref{t:qeta}, we obtain the following
$q$-analog of Theorem~\ref{t:thetavw}.

\begin{thm}\label{t:qthetavw}
Let $v, w \in \sn$ \avoidp{} and satisfy $v \sim w$, 
and let $(X, \STAT)$
be a property of $F$-tableaux and a statistic on $F$-tableaux 
which depend only upon the poset $P(F)$.
Then $F_v$-tableaux 
%$U$ 
and $F_w$-tableaux 
%$U$ 
having property $X$ and 
satisfying $\STAT(U) = k$ are in bijective correspondence.  Moreover,
%Then 
for each $\hnq$-trace $\theta_q$ we have 
\begin{equation*}
%\label{eq:thetacvcw}
\theta_q(\qev C'_v(q)) = \theta_q(\qew C'_w(q)).
%= \theta_q(q_{e,w_0ww_0} C'_{w_0ww_0}(q)).
\end{equation*}
%Furthermore, there exists a $312$-avoiding permutation 
%$u \in \sn$ which satisfies
%\begin{equation}\label{eq:thetacvcu}
%\theta_q(\qev C'_v(q)) = \theta_q(\qeu C'_u(q)).
%\end{equation}
\end{thm}
\begin{proof}
Since the pair $(X, \STAT)$ depends 
only upon the poset $P(F_v) \cong P(F_w)$,
we have a bijection between the sets of 
$F_v$-tableaux 
%$U$
%which depends only upon the poset is also a 
and $F_w$-tableaux 
%$U$ 
having property $X$ and satisfying $\STAT(U) = k$.
In particular, we have 
$\eta_q^\lambda(\qev C'_v(q)) = \eta_q^\lambda(\qew C'_w(q))$, 
and since $\{ \eta_q^\lambda \,|\, \lambda \vdash n \}$ is a basis of the
space of $\hnq$-traces, we have
%also have 
%the equality
$\theta_q(\qev C'_v(q)) = \theta_q(\qew C'_w(q))$ for all $\hnq$-traces 
$\theta_q$ as well.
%($\star$ Do we have to throw this out?  Use Shareshian-Wachs?
%Express $C'_{w_0ww_0}(q))$ as something like $\wT_{w_0} C'_w(q) \wT_{w_0}^{-1}$?)
%Now consider the permutation $w_0ww_0$.  The poset $P(F_{w_0ww_0})$ is isomorphic
%to the dual of $P(F_w)$.
%Let $\pi'' = (\pi''_1, \dotsc, \pi''_n)$ be the unique path family of type $e$
%which covers $F_{w_0ww_0}$.  Now 
%use the $F_v$-tableau $U$ to create a tableau $U''$ by replacing each
%entry $\pi_i$ of $U$ by $\pi''_{n+1-i}$, and reversing the order of paths
%from bottom to top in each column.  It is easy to see that $U''$ is
%a column-strict $F_{w_0ww_0}$-tableau of type $e$ and shape $\lambda$.
%Furthermore, the map $U \mapsto U''$ preserves inversions, 
%since $\pi_j$ appears to the left of $\pi_i$ in $U$ if and only if
%$\pi''_{n+1-j}$ appears to the left of $\pi_{n+1-i}$ in $U''$.
%Equation (\ref{eq:thetacvcu}) now follows 
%($\star$ if we know that inversions and intersecting noninversions are equally distributed on the set of column-strict $F_w$-tableaux).
\end{proof}
Let $\theta_q$ be an $\hnq$-trace.
If the posets $P(F_v)$, $P(F_w)$ of two zig-zag networks are dual,
rather than isomorphic, we still have
$\theta_q(\qev C'_v(q)) = \theta_q(\qew C'_w(q))$.  
In this case $v$ and $w$ satisfy $v \sim w_0ww_0$ and $\ell(v) = \ell(w)$, 
where $w_0$ is the longest element of $\sn$. 
Since natural basis elements of $\hnq$ satisfy 
$T_{w_0}T_w T_{w_0}^{-1} = T_{w_0ww_0}$, and since any $\hnq$-trace $\theta_q$
satisfies $\theta_q(gh) = \theta_q(hg)$ for all $h, g \in \hnq$,
%and since 
we have that the equations
\begin{equation*}
%\label{eq:qdualposeteq}
\theta_q(q_{e,w_0ww_0} C'_{w_0ww_0}(q)) = 
\theta_q(\qew T_{w_0} C'_w(q) T_{w_0}^{-1}) = 
\theta_q(\qew C'_w(q))
%\theta_q(q_{e,w^{-1}} C'_{w^{-1}}(q)).
\end{equation*}
hold for all $w \in \sn$.
%\begin{equation}\label{eq:dualposeteq}
%\theta(C'_{w_0ww_0}(1)) = \theta(w_0 C'_w(1) w_0) = \theta(C'_w(1))
%\end{equation}
%hold for all $w \in \sn$.
%It is easy to show that we 
%As remarked earlier, we have $w \sim w^{-1}$.
%In analogy to 
%As in (\ref{eq:dualposeteq}), we have
%$\theta_q(\qew C'_v(q)) = \theta_q(\qew C'_w(q))$ when the posets $P(F_v)$, 
%$P(F_w)$ are dual.
% $v \sim w$: for
%it is possible to have 
%$\theta_q(q_{e,w_0ww_0} C'_{w_0ww_0}(q)) = 
%\theta_q(\qew T_{w_0} C'_w(q) T_{w_0}^{-1}) = 
%\theta_q(\qew C'_w(q))$,
%In this case we have $v \sim w_0ww_0$ and the equations
%, we have
%: both of these evaluations are equal to
%$\theta_q(\qew C'_v(q)) = \theta_q(\qew C'_w(q))$ without $v \sim w$: for
%all $w \in \sn$ we have
% is not necessary
%we have pairs $(w, w_0ww_0)$
%it is easy to show that while
%show that reflection
%of $F_w$ in a vertical ($F_{w^{-1}}$) or horizontal ($F_{w_0ww_0}$) line
%yields the equal evaluations
%we have
%We also have 
%where $w_0$ is the ``longest'' permutation in $\sn$, having one-line notation
%$n(n-1) \cdots 1$. While the posets $P(F_{w_0ww_0})$, $P(F_w)$ are dual,
%rather than isomorphic, it is easy to show that we have 
%Since $\ell(w_0ww_0)= \ell(w)$, both of these expressions are equal to
%we have
%\begin{equation*}
%Specifically, both expressions are equal to
%we have $\theta_q(q_{e,w_0ww_0} C'_{w_0ww_0}(q)) = 
%$\theta_q(\qew T_{w_0} C'_w(q) T_{w_0}^{-1})$.
% = \theta_q(\qew C'_w(q))$.
%\end{equation*}
%since $w_0 = w_0^{-1}$.
%($\star$ If $P(F_v)$ is isomorphic to the dual of $P(F_w)$,
%does this imply the same equality?)

%$\star$ Include Sam's special case.

\section{Interpretation of $\epsilon_q^\lambda(\qew C'_w(q))$}\label{s:hnqinterpe}
%\subsection{Interpretation of $\epsilon_q^\lambda(\qew C'_w(q))$
%and $\chi_q^\lambda(\qew C'_w(q))$}
%Now let us consider a combinatorial interpretation for
%$\epsilon_q^\lambda(\qew C'_w(q))$ 
%where $w \in \sn$ avoids the pattern $312$.
%-avoiding permutation.
% corresponding
%to unit interval order $P$ and descending star network $\hat F$.

%($\star$ Check the results in this subsection
%to ensure that they hold for \pavoiding permutations.)

Let $w \in \sn$ \avoidp\ and
%Letting 
let $B$ be the path matrix of $F_w$.  Following the computations
of Equations (\ref{eq:hqlambeta}) -- (\ref{eq:hqlambetaalt}),
we have
\begin{equation}\label{eq:eqlambepsilon}
\begin{aligned}
\epsilon_q^\lambda(\qew C'_w(q)) 
= \sigma_B(\imm{\epsilon_q^\lambda}(x)) 
&= \sum_{(I_1,\dotsc,I_r)} \sigma_B(\qdet(x_{I_1,I_1}) \cdots \qdet(x_{I_r,I_r})) \\
&= \sum_{u \in \slambdamin} \sum_{y \in u^{-1}\slambda u} 
(-1)^{\ell(uy)-\ell(u)}\sigma_B (q_{u,uy}^{-1} x^{u,uy}),
\end{aligned}
\end{equation}
where the first sum is over all ordered set partitions $(I_1,\dotsc, I_r)$
of $[n]$ of type $\lambda$.
Let us therefore consider evaluations of the form 
%\begin{equation*}
$\sigma_B (\qiuv x^{u,v})$.
%\end{equation*}

To combinatorially interpret these evaluations, 
%we consider 
%path families and certain orderings of these in tableaux.
let $\pi = (\pi_1, \dotsc, \pi_n)$ be a path family 
(of arbitrary type)
which covers a zig-zag
%descending star 
network $F$ and
define $U(u,\pi,\lambda)$ to be the $\pi$-tableau of shape $\lambda$
containing $\pi$ in the order $\pi_{u_1}, \dotsc, \pi_{u_n}$.
%which contains the paths of $\pi$ in the order $\pi_{u_1}, \dotsc, \pi_{u_n}$.
That is, 
%the $j$th row of 
$U(u, \pi, \lambda)_j$
contains the $\lambda_j$ paths whose indices are
\begin{equation*}
u_{\lambda_1 + \cdots + \lambda_{j-1}+1}, \dotsc, u_{\lambda_1 + \cdots + \lambda_j}.
\end{equation*}
Clearly 
%the left tableau of 
$L(U(u,\pi,\lambda))$ 
contains the numbers 
$u_1, \dotsc, u_{\lambda_1}$ in row 1, 
$u_{\lambda_1 + 1}, \dotsc, u_{\lambda_1+\lambda_2}$ in row 2, etc. 
If $R(U(u,\pi,\lambda))$ 
%the right tableau 
contains the numbers
$v_1, \dotsc, v_{\lambda_1}$ in row 1, 
$v_{\lambda_1 + 1}, \dotsc, v_{\lambda_1+\lambda_2}$ in row 2, etc.,
then $\pi$ has type $u^{-1}v$.
In terms of this notation, our earlier tableau $U(u,\pi)$ 
defined after (\ref{eq:hqlambetaalt}) is equal to 
$U(u,\pi,(n))$.
%If $s_i$ is a left descent for $u$, then right inversions in $U(u,\pi)$
%and $U(s_iu,\pi)$ are related as follows.
If $u \in \slambdamin$ corresponds to $(I_1, \dotsc, I_r)$ as in
(\ref{eq:subwordsofu}) -- (\ref{eq:hqlambetaalt}),
%($\star$ or something like that?), 
then the path indices in row $j$ of $U(u,\pi,\lambda)$
are simply the elements of $I_j$, in increasing order. 


\begin{prop}\label{p:sigaqiuvxuv}
Let $w \in \sn$ \avoidp, 
%the pattern $312$
%Let $\hat F$ be the descending star network 
%corresponding to the $312$-avoiding permutation $w$ in $\sn$, 
and let $B$ be the path matrix of $F_w$. 
%of the unit interval order corresponding to $\hat F$.
Fix $\lambda \vdash n$, $u \in \slambdamin$, and $y \in u^{-1}\slambda u$.
%define $I = I(u)$ as in $v$ in $\sn$.  
Then we have
\begin{equation*}
%\label{eq:sigaqiuvxuv}
\sigma_B (q_{u,uy}^{-1} x^{u,uy}) = 
\begin{cases}
q^{\rinv(U(u,\pi,\lambda)^\tr)} 
&\text{if there exists a path family $\pi$ of type $y$ 
covering $F_w$},\\
0 &\text{otherwise}.
\end{cases}
\end{equation*}
\end{prop}
\begin{proof}
By Proposition~\ref{p:sigaquvxuv} we have
\begin{equation*}
\begin{aligned}
\sigma_B (q_{u,uy}^{-1} x^{u,uy}) &= q_{u,uy}^{-2} \sigma_B (q_{u,uy} x^{u,uy}) \\
&= \begin{cases} 
q^{\ell(u)-\ell(uy)} q^{\rinv(U(u,\pi,(n)))} 
&\text{if some path family of type $y$ covers $\hat F$,}\\
0 &\text{otherwise.}
\end{cases}
\end{aligned}
\end{equation*}

Letting $V = U(u,\pi,\lambda)$ 
and using (\ref{eq:rinvudotsu}),
we may rewrite the above exponent of $q$ as
$\rinv(V_1 \circ \cdots \circ V_r) + \ell(u) - \ell(uy)$.
By (\ref{eq:invdecomp}), this is
%, or
% \\
\begin{equation*}
%\begin{gathered}
%= 
\rinv(V_1) + \cdots + \rinv(V_r) + \rinv(V^\tr) - (\ell(uy) - \ell(u)).
%\end{gathered}
\end{equation*}
%$\star$ Be careful.  We don't have $\ell(uy) = \ell(u) + \ell(y)$ here.
We claim that this expression reduces further to $\rinv(V^\tr)$.
To see this, 
recall that the tableaux $L(V_1) \circ \cdots \circ L(V_r)$ 
and $R(V_1) \circ \cdots \circ R(V_r)$ contain
the one-line notations of $u$ and $uy$, respectively.
Since $u$ belongs to $\slambdamin$ and each tableau $R(V_j)$ is a permutation
of the corresponding tableau $L(V_j)$,
% are equal
we have 
%that 
%$\ell(u)$ is equal to the number of pairs $(k,k')$, $k > k'$, with 
%$k$ appearing in an earlier row of $L(V)$ than $k'$.
\begin{equation*}
\begin{gathered}
\begin{aligned}
\ell(u) 
&= \# \{ (k,k') \,|\, k > k', 
\text{ $k$ in an earlier row of $L(V)$ than $k'$} \}\\
&= \# \{ (k,k') \,|\, k > k', 
\text{ $k$ in an earlier row of $R(V)$ than $k'$} \}.
\end{aligned}\\
\begin{aligned}
\ell(uy) &= \sum_{j=1}^r \inv(R(V_j)) + 
\# \{ (k,k') \,|\, k > k', 
\text{ $k$ in an earlier row of $R(V)$ than $k'$} \}\\
&= \sum_{j=1}^r \inv(R(V_j)) + \ell(u).
\end{aligned}
\end{gathered}
\end{equation*}

%Similarly,
%%Now observe that 
%the tableau
%$R(V_1) \circ \cdots \circ R(V_r)$ contains 
%the one-line notation of the permutation $uy$.
%Thus, $\ell(uy)$ is equal to $\inv(R(V_1)) + \cdots + \inv(R(V_r))$ plus
%the number of pairs $(k,k')$, $k > k'$, with $k$ appearing in an earlier
%row of $R(V)$ than $k'$.
%Since $V$ is a row-closed tableau, this is just
%\begin{equation*}
%\inv(R(V_1)) + \cdots + \inv(R(V_r)) + \ell(u).
%\end{equation*}
%observe that the one-line notation of the permutation $u$
%is the concatenation $L(V_1) \circ \cdots \circ L(V_r)$.  
%since $u$ belongs to $\Wlem$, we have that 
%\begin{equation*}
%\ell(u) = \inv(L(V_1) \circ \cdots \circ ber $\ell(u)$ is e
%\end{equation*}

Now fix a row $V_j$ of $V$ and consider right inversions in $V_j$.
Let $\pi_i$, $\pi_{i'}$ be two paths in this row, with $\pi_i$ appearing first.
Since $u$ belongs to $\slambdamin$, we have $i < i'$.
Let $k$ and $k'$ be the corresponding sink indices.
If we have $k > k'$, then the paths cross and form a right inversion in $V_j$.
%Moreover, the pair $(k,k')$ is an inversion in the permutation $uy$.
On the other hand, if we have $k < k'$, 
then the paths do not form a right inversion in $V_j$, even if they intersect.
%and the pair $(k,k')$ is not an inversion in the permutation $uy$.  
Thus we have $\rinv(V_j) = \inv(R(V_j))$ for all $j$ and
\begin{equation}\label{eq:luylu}
\rinv(V_1) + \cdots + \rinv(V_r) = \ell(uy) - \ell(u),
\end{equation}
as desired.
%Now consider the total number of inversions in the permutation $uy$.
\end{proof}


While the final sum in (\ref{eq:eqlambepsilon}) has signs,
we will use a sign-reversing involution to cancel some of the
the terms there, and to obtain a signless sum more amenable to 
combinatorial interpretation.
Fix a partition $\lambda = (\lambda_1, \dotsc, \lambda_r) \vdash n$,
an ordered set partition $I$ of $[n]$ of type $\lambda$, 
and a zig-zag
%descending star 
network $F$,
%and let $\mathcal T_\lambda = \mathcal T_\lambda(F)$ 
and let $\mathcal T_I = \mathcal T_I(F)$ be 
the set of all row-closed, left row-strict $F$-tableaux $U$ of shape $\lambda$
such that $L(U)_j = I_j$ (as sets) for $j = 1,\dotsc,r$.
Observe that all row-strict $F$-tableaux of type $e$ 
%belong to $\mathcal T_\lambda$.
satisfying $L(U)_j = I_j$ (as sets) for $j = 1,\dotsc,r$
belong to $\mathcal T_I$.
%($\star$ Start to use {\em canonical} to emphasize that in a row-closed 
%tableaux $U$, not only is $R(U_j)$ equal to a permutation of $L(U_j)$ 
%for all $j$, but also the left tableau is row-strict.)
Let us define an involution
\begin{equation*}
%\label{eq:zeta}
%\zeta: \mathcal T_\lambda \rightarrow \mathcal T_\lambda
\zeta: \mathcal T_I \rightarrow \mathcal T_I
\end{equation*}
as follows.  
%($\star$ This definition isn't quite right.  Fix it.)  (Yes it is?)
\begin{enumerate}
\item If $U$ is a row-strict tableau of type $e$, then define $\zeta(U) = U$.
\item Otherwise, 
  \begin{enumerate}
  \item Let $i$ be the least
%greatest 
index such that $U_i$ is not row-strict.
  \item 
Let $(j,j')$ be the 
%right-to-left lexicographically greatest 
lexicographically least
pair of indices in $L(U)_i$ such that 
%$j < j'$, path $\pi_j$ appears to the left of path $\pi_{j'}$
$\pi_j$ and $\pi_{j'}$ 
%appear in $U_i$, and the two paths 
intersect.
%Let $(j,j')$ be the lexicographically greatest pair of indices 
%such that $j < j'$, path $\pi_j$ appears to the left of path $\pi_{j'}$
%in $U_i$, and the two paths intersect.
  \item Let $(k,k')$ be the sink indices of paths $\pi_j$ and $\pi_{j'}$, respectively.
  \item Define $\zeta(U)$ to be the tableau obtained from $U$ by replacing
$\pi_j$ and $\pi_{j'}$ by the unique paths in $F$ from
source $j$ to sink $k'$ and source $j'$ to sink $k$.
  \end{enumerate}
\end{enumerate}

\begin{prop}\label{p:zeta}
The involution $\zeta$ satisfies 
%\begin{enumerate}
%\item 
$\rinv(\zeta(U)^\tr) = \rinv(U^\tr)$.
%\item $\rinv(\zeta(U)) \in \{ \rinv(U) \pm 1 \}$.
%\end{enumerate}
\end{prop}
\begin{proof}
%(1.) 
For each $F$-tableau $U \in \mathcal T_I$ 
%$U \in \mathcal T_\lambda$ 
satisfying $\zeta(U) = U$, the 
claimed equality is obvious.
Now let $U$ be an $F$-tableau not fixed by $\zeta$. Define the indices
$i$, $j$, $j'$, $k$, $k'$ 
and paths 
$\pi_j$, $\pi_{j'}$ as in the definition of $\zeta$,
and let $\pi'_j$, $\pi'_{j'}$ be the two new paths created in the final step
of the definition of $\zeta$.
Since the tableaux $U^\tr$ and $\zeta(U)^\tr$ agree everywhere except in the
two cells in column $i$ containing the paths 
\begin{equation}\label{eq:4paths}
\pi_j, \quad \pi_{j'}, \quad \pi'_j, \quad \pi'_{j'},
\end{equation}
%(\ref{eq:4paths}), 
it is clear 
%and let $\pi_h$ be a path appearing in $U \ssm U_i$
%%$U_1, \dotsc, U_{i-1}$ 
%and having sink index $\ell$.  
%It is clear that neither 
%$\pi_j$, $\pi_{j'}$ nor 
%$\pi'_j$, $\pi'_{j'}$ form inv 
that the right inversions of these tableaux are equal except possibly for 
inversions involving a path $\pi_h$ in a column other than $i$ 
and one of the paths (\ref{eq:4paths}).
We claim that these remaining right inversions in $U^\tr$ and $\zeta(U)^\tr$
correspond bijectively.  In particular, we have
\begin{enumerate}[(a)]
\item $\pi_h$ forms a right inversion with $\pi_j$ in $U^\tr$
if and only if
it forms a right inversion with $\pi'_{j'}$ in $\zeta(U)^\tr$.
%$\{ \pi_h, \pi'_{j'} \}$ form a right inversion in $\zeta(U)^\tr$.
%and
\item $\pi_h$ forms a right inversion with $\pi_{j'}$ in $U^\tr$
if and only if
it forms a right inversion with $\pi'_{j}$ in $\zeta(U)^\tr$.
%\item $\{ \pi_h, \pi_{j'} \}$ form a right inversion in $U^\tr$
%if and only if
%$\{ \pi_h, \pi'_j \}$ form a right inversion in $\zeta(U)^\tr$.
\end{enumerate}

%the set of right inversions in $U^\tr$ not involving the paths (\ref{eq:4paths})
%which involve $\pi_h$
%and paths other than $\pi_j$, $\pi_{j'}$ 
%is equal to the set of right inversions in $\zeta(U)^\tr$ 
%not involving these paths.
%which involve $\pi_h$
%and paths other than $\pi'_j$, $\pi'_{j'}$.

To see this, observe that $j < j'$, 
and consider the intersection of $\pi_h$ with $\pi_j$ and $\pi_{j'}.$
%Since $U$ is left row-strict we have $j < j'$.  
%Since $F$ is a descending star network,
%the subpath of $\pi_j$
%preceding $\pi_j \cap \pi_{j'}$ lies entirely below the subpath of $\pi_{j'}$
%preceding this intersection, and
%the subpath of $\pi_j$ or $\pi_{j'}$ following this intersection 
%and terminating at sink $\min \{k,k'\}$ 
%lies entirely below
%the subpath 
%following this intersection 
%and terminating at sink $\max \{k,k'\}$.
By Lemma~\ref{l:dsnintersect} 
%it follows that 
%Since $F$ is a descending star network $\pi_h$ intersects the other paths
%in one of four ways:
we have four cases: 
%($\star$ Explain in greater detail.)
\begin{enumerate}
\item $\pi_h$ intersects both $\pi_j$ and $\pi_{j'}$.
\item $\pi_h$ intersects only $\pi_{j'}$.
\item $\pi_h$ intersects only the path whose sink is $\min(k,k')$.
\item $\pi_h$ intersects neither $\pi_j$ nor $\pi_{j'}$.
\end{enumerate}
%It follows that the intersection of $\pi_h$ with $\pi'_j$ and 
Now considering the intersections of $\pi_h$ with $\pi'_j$ and $\pi'_{j'}$,
we see that the above cases
%These cases 
imply respectively that $\pi_h$ intersects
both $\pi'_j$ and $\pi'_{j'}$,
%In the first case $\pi_h$ intersects both $\pi'_j$ and $\pi'_{j'}$;
%and the equivalences are true.
%Thus we have
%In the second case, $\pi_h$ intersects 
only $\pi'_{j'}$ (not $\pi'_j$),
%and again the equivalences are true.
% we have the first equivalence above.
%In the third case, $\pi_h$ intersects either $\pi'_j$ or $\pi'_{j'}$,
only the path in $\{\pi'_j, \pi'_{j'} \}$ whose
%whichever has 
sink index is $\min(k,k')$, 
%and again 
%the equivalences are true.
%(a) and (b).
%In the fourth case, $\pi_h$ intersects 
and neither $\pi'_j$ nor $\pi'_{j'}$.
In all cases, the equivalences (a) and (b) are true.
%and thus forms no inversions with the 
%paths (\ref{eq:4paths}).
%In the first case $\pi_h$ intersects both $\pi'_j$ and $\pi'_{j'}$,
%and the equivalences are true.
%In the second case, $\pi_h$ intersects $\pi'_{j'}$ but not $\pi'_j$,
%and again the equivalences are true.
%% we have the first equivalence above.
%In the third case, $\pi_h$ intersects either $\pi'_j$ or $\pi'_{j'}$,
%whichever has sink index $\min(k,k')$, and again 
%the equivalences are true.
%%(a) and (b).
%In the fourth case, $\pi_h$ intersects neither $\pi'_j$ nor $\pi'_{j'}$
%and thus forms no inversions with the 
%paths (\ref{eq:4paths}).
\end{proof}

%It is easy to see that right inversions within a fixed row of 
Since the tableaux $U$ and $\zeta(U)$ agree except in one row,
%the right inversions in all but that row are equal, unless that row is 
we have the following.

\begin{prop}\label{p:obv}
Let $U \in \mathcal T_I$ be a tableau 
%Let $U \in \mathcal T_\lambda$ be a tableau 
not fixed by $\zeta$ and let $i$ be the index
satisfying $U_i \neq \zeta(U)_i$.  Then we have
\begin{equation*}
|\rinv(\zeta(U)_j) - \rinv(U_j)| = 
\begin{cases}
1 &\text{if $j = i$},\\
0 &\text{otherwise}.
\end{cases}
\end{equation*}
\end{prop}
\begin{proof}
Obvious.
\end{proof}

%two paths in 
%$U_i$ intersect, and let $, \dotsc$
% (some row of $U$ has two intersecting paths), let $i$
%be the greatest such row and let it, and let $(j,j')$ be the lexicographically 
%greatest triple such that in row $i$,
%the paths in positions $j$ and $j'$ intersect.
%\begin{enumerate} 

Now we have the following $q$-analog of Theorem~\ref{t:sncfinterp} (i).

\begin{thm}\label{t:qepsilon}
Let $w \in \sn$ \avoidp.
% the pattern $312$
%and let $B$ be the path matrix of $F_w$. 
%-avoiding permutation corresponding
%to unit interval order $P$ and descending star network $\hat F$.
Then for 
%$\lambda = (\lambda_1, \dotsc, \lambda_r) \vdash n$ 
$\lambda \vdash n$ 
we have
\begin{equation*}
%\label{eq:epsilonmain}
\epsilon_q^\lambda(\qew C'_w(q)) = \sum_U q^{\inv(U)},
\end{equation*}
where the sum is over all column-strict $F_w$-tableaux 
%$U$ 
of 
type $e$
and 
shape $\lambda^\tr$.
\end{thm}
\begin{proof}
Let $B$ be the path matrix of $F_w$ and let $(I_1,\dotsc,I_r)$ be a set
partition of $[n]$ of type $\lambda$.
%satisfying $|I_j| = \lambda_j$ for $j = 1,\dotsc,r$.
By 
%the comment following 
(\ref{eq:subwordsofu}) -- (\ref{eq:hqlambetaalt}),
%Proposition~\ref{p:qimmchar} and 
%(\ref{eq:hqlambetaalt}),
there is a permutation $u \in \slambdamin$ 
corresponding to $(I_1,\dotsc,I_r)$ such that 
we have
\begin{equation*}
%\eta_q^\lambda(\qew C'_w(q)) = \sum_{u \in \slambdamin} 
\sigma_B(\qdet(x_{I_1,I_1}) \cdots \qdet(x_{I_r,I_r})) = 
\sum_{y \in u^{-1}\slambda u} (-1)^{\ell(uy)-\ell(u)} \sigma_B ( q_{u,uy}^{-1} x^{u, uy} ).
\end{equation*}
%Now consider one permutation $u \in \slambdamin$ 
%and define the ordered set partition 
%\begin{equation*}
%I = I(u) = (I_1, \dotsc, I_r)
%\end{equation*}
%as in (\ref{eq:Iudef}).
By Proposition~\ref{p:sigaqiuvxuv}
this is equal to
%e inner sum may be rewritten as
%we have
\begin{equation}\label{eq:ypiinvuupi}
\sum_{(y,\pi)} (-1)^{\ell(uy)-\ell(u)} q^{\rinv(U(u,\pi,\lambda)^\tr)},
\end{equation}
where the 
%last 
%inner sum now 
sum is over pairs $(y,\pi)$ such that 
$y \in u^{-1}\slambda u$ and 
$\pi$ is a path family of type $y$ which covers $F_w$.
If such a path family exists, it is necessarily unique.  
Thus as $y$ varies over $u^{-1}\slambda u$
%As $u$ varies over $\slambdamin$ and
%.  Letting 
%$(y,\pi) = (y, \pi(y))$ varies as above, 
we have that
%the tableau 
$U(u, \pi,\lambda)$
% = U(u,\pi(y))$ 
varies over all 
tableaux in $\mathcal T_I$. 
%($\star$ Why?)
Thus by (\ref{eq:luylu}) this sum is equal to
\begin{equation}\label{eq:Vprecancel}
%\begin{aligned}
%\epsilon_q^\lambda(\qew C'_w(q)) &=
%\sum_{u \in \slambdamin} \sum_{(y,\pi)} (-1)^{\ell(uy)-\ell(u)} q^{\rinv(U(u,\pi,\lambda)^\tr)}\\
%&= 
\sum_{V \in \mathcal T_I} (-1)^{\rinv(V_1) + \cdots + \rinv(V_r)} q^{\rinv(V^\tr)}.
%\end{aligned}
\end{equation}

Now consider a tableau $V \in \mathcal T_I$ which satisfies 
$\zeta(V) \neq V$. By Propositions~\ref{p:zeta} -- \ref{p:obv},
the term of the above sum corresponding to the tableau $\zeta(V)$ is
\begin{equation*}
(-1)^{\rinv(\zeta(V)_1) + \cdots + \rinv(\zeta(V)_r)} q^{\rinv(\zeta(V)^\tr)}
= -(-1)^{\rinv(V_1) + \cdots + \rinv(V_r)} q^{\rinv(V^\tr)}.
\end{equation*}
Thus all terms corresponding to tableaux $V$ and $\zeta(V) \neq V$
cancel one another in the sum (\ref{eq:Vprecancel}), leaving terms
only for the tableaux $V \in \mathcal T_I$ 
which are fixed by $\zeta$, i.e., the row-strict tableaux.  
For these tableaux we have
\begin{equation*}
\rinv(V_1) = \cdots = \rinv(V_r) = 0, \qquad
\rinv(V^\tr) = \inv(V^\tr).
\end{equation*}
Furthermore, $V$ is row-strict of shape $\lambda$ if and only if
$V^\tr$ is column-strict of shape $\lambda^\tr$.  Thus we may again rewrite
(\ref{eq:ypiinvuupi}) as
\begin{equation*}
\sum_U q^{\inv(U)},
\end{equation*}
where the sum is over all column-strict $F_w$-tableaux of shape $\lambda^\tr$
satisfying $U_j = I_j$ (as sets).
%($\star$ What equation relating \rinv and \inv?)
%These are row-strict $F_w$-tableaux of shape $\lambda$, 
%as desired.
%($\star$ Does this sign reversing involution work
%if we restrict our attention to one term
%$\qdet(x_{I_1,I_1}) \cdots \qdet(x_{I_r,I_r})$?)
%($\star$ Finish this proof.)
%Thus the expression
%over all $u \in \Wlem$ of the second sum 
%in (\ref{eq:ypiinvuupi})
%(\ref{eq:ypirinvuupi})
%By Lemma ?, this 
%may be rewritten as
%\begin{equation}\label{eq:rinvudotsu}
%\sum_U q^{\rinv(U_1 \circ \cdots \circ U_r)},
%\end{equation}
%where this last sum is over all 
%row-closed, left row-strict 
%$F_w$-tableaux $U$ of shape $\lambda$
%for which path indices of $U_j$ are $I_j$ for $j=1,\dotsc,r$.
Summing over ordered set partitions and using (\ref{eq:eqlambepsilon}),
we have the desired result.
%which satisfy $L(U) = (u_1, \dotsc, u_n)$.
\end{proof}

%($\star$ Do an example.)

For example, consider the descending star network $F_{3421}$ in
(\ref{eq:F3421}).
It is easy to verify that 
there are exactly two column-strict
$F_{3421}$-tableaux of type $e$ and shape $31$:
\begin{equation}\label{eq:epsilontabs31}
\raisebox{-6mm}{\includegraphics[height=12mm]{tableaux/31_4123}}\ ,\quad
\raisebox{-6mm}{\includegraphics[height=12mm]{tableaux/31_4132}}\ ,
%\quad
%\raisebox{-6mm}{\includegraphics[height=12mm]{tableaux/31_33122441}}\ ,\quad
%\raisebox{-6mm}{\includegraphics[height=12mm]{tableaux/31_22133441}}\ ,
\end{equation}
where $\pi_i$ is the unique path from source $i$ to sink $i$.
These tableaux 
have $2$ and $3$ inversions, respectively.
%$U$ of shape $31$
%and if we construct from each tableau $U$ 
%yield tableaux $U_1 \circ U_2$ of shape $4$,
%\begin{equation}\label{eq:etatabs4}
%\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_12213344}}\ ,\quad
%\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_13213244}}\ ,\quad
%\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_12244133}}\ ,\quad
%\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_13344122}}\ ,
%\end{equation}
%which have $1$, $2$, $3$, and $4$ right inversions, respectively.
Since $31^\tr = 211$,  
%they
the tableaux together give 
%contribute $q + q^2 + q^3 + q^4$ to 
$\epsilon_q^{211}(q_{e,3421} C'_{3421}(q)) = q^2 + q^3$.
It is clear that all twenty-four $F_{3421}$-tableaux of type $e$ and shape $4$ 
are column-strict.  Counting inversions in these tableaux gives
$\epsilon_q^{1111}(q_{e,3421} C'_{3421}(q)) = 1 + 3q + 8q^2 + 8q^3 + 3q^4 + q^5$.
It is also clear that there are no $F_{3421}$-tableaux of type $e$ and 
shapes $22$, $211$, or $1111$. Thus we have
$\epsilon_q^{22}(q_{e,3421} C'_{3421}(q)) = 
\epsilon_q^{31}(q_{e,3421} C'_{3421}(q)) = 
\epsilon_q^{4}(q_{e,3421} C'_{3421}(q)) = 0$.

%\begin{cor}
%Let $w \in \sn$ \avoidp, 
%let $B$ be the path matrix of $F_w$,
%and let $(I_1,\dotsc,I_r)$ be an ordered set partition of $[n]$.
%Then we have
%\begin{equation*}
%\sigma_B(\qdet(x_{I_1,I_1}) \cdots \qdet(x_{I_r,I_r})) = 
%\sum_U q^{\inv(U)},
%\end{equation*}
%where the sum is over all column-strict 
%$F_w$-tableaux $U$ of shape $(|I_1|, \dotsc, |I_r|)^\tr$
%and type $e$
%for which path indices in $U_j^\tr$ are 
%$\lambda_1 + \cdots + \lambda_{j-1}+1, \dotsc, \lambda_1 + \cdots + \lambda_j$,
%for $j = 1,\dotsc,r$.
%\end{cor}
%($\star$ Incorporate this into proof of theorem?)
%Haiman~\cite{HaimanHecke} has shown
%that we have $\chi_q^\lambda(\qew C'_w(q)) \in \mathbb N[q]$ 
%for all $\lambda \vdash n$ and all $w \in \sn$.
%($\star$ Mention symmetry and unimodality too.)
%($\star$ Where does this comment belong? Not here.)

Expanding $\epsilon_q^\lambda$ in terms of irreducible characters
and Kostka numbers, $\epsilon_q^\lambda = \sum K_{\mu^\tr,\lambda} \chi_q^\mu$,
%where $\{ K_{\mu,\lambda} \,|\, \mu, \lambda \vdash n \}$ are the Kostka numbers,
and using Haiman's result~\cite[Lem\,1.1]{HaimanHecke},
we have that $\epsilon_q^\lambda( \qew C'_w(q))$ is symmetric and unimodal
about $\qew$ for all $w \in \sn$.
In the case that $w$ \avoidsp, it would be interesting to explain
this phenomenon combinatorially in terms of Theorem~\ref{t:qepsilon}.

\begin{cor}\label{c:edeck}
Fix 
$\lambda = (\lambda_1, \dotsc, \lambda_r) \vdash n$
%$\lambda \vdash n$
and $w \in \sn$ \avoidingp.  If $w_1 \cdots w_n$ has 
a decreasing subsequence of length greater than 
$r$,
%$\lambda_1^{\tr}$, 
then 
%we have 
$\epsilon_q^\lambda(\qew C'_w(q)) = 0$.
\end{cor}
\begin{proof}
Let $(\pi_1,\dotsc,\pi_n)$ be the unique path family of type $e$ which covers
$F_w$.  By Theorem~\ref{t:kstar} 
there exist $r+1$ paths $\pi_i, \dotsc, \pi_{i+r}$
which share a vertex.  No two of these can appear together in a column
of a column-strict $F_w$-tableau. Thus no such tableau has
%there is no column-strict $F_w$-tableau of 
shape $\lambda^\tr$.
\end{proof}

More generally, 
it is known that we have $\epsilon_q^\lambda(\qew C'_w(q)) = 0$
unless $\lambda \leq \sh(w)$ in the majorization order, where
$\sh(w)$ is the partition associated to $w$ 
by the Robinson-Schensted row insertion algorithm. 
(See, e.g., \cite[Prop.\,4.1\,(3)]{HaimanHecke}.)
This implies Corollary~\ref{c:edeck}, since a decreasing subsequence
of length greater than $r$
%$\lambda_1^{\tr}$ 
in $w_1 \dotsc w_n$ implies that
we have $\lambda \nleq \sh(w)$.

\section{\ntnsp Chromatic symmetric and 
quasisymmetric functions}\label{s:chromsf}
%\section{Generating functions for $\theta(C'_w(1))$, $\theta_q(\qew C'_w(q))$
%when $w$ is fixed}\label{s:chromsf}

%$\star$ Rewrite with original definitions of chromatic (quasi-) symmetric
%functions and put my versions in the proofs of the theorems here.

The evaluations of $\sn$-class functions and $\hnq$ traces
at Kazhdan-Lusztig basis elements
%For $\theta^\lambda$ an $\sn$-class function
%and $\theta_q^\lambda$ an $\hnq$-trace,
%the evaluations $\theta^\lambda(C'_w(1))$ and $\theta_q^\lambda(\qew C'_w(q))$
are closely related to certain symmetric and quasisymmetric functions
defined by 
%Stanley-Stembridge~\cite{StanSymm} \cite{StanStemIJT},
Stanley~\cite{StanSymm}
and Shareshian and Wachs~\cite{SWachsChromQ}.

Let 
$\Lambda_n$ 
%\smash{\defeq}$ 
%\mathrm{span}_{\mathbb Z}\{ m_\lambda \,|\, \lambda \vdash n\}$ 
%$\Lambda_n = \mathrm{span}_{\mathbb Z}\{ m_\lambda \,|\, \lambda\vdash n\}$ 
be the $\mathbb Z$-module of homogeneous 
%integer span of space of 
degree $n$ symmetric functions.
%For $P$ an $n$-element poset,
%Stanley and Stembridge~\cite{StanSymm}, \cite{StanStemIJT}
In \cite{StanSymm} Stanley
defined certain {\em chromatic symmetric functions}
$\{ X_G \,|\, G \text{ a simple graph on $n$ vertices}\,\}$ in $\Lambda_n$
and studied expansions of these in various bases of $\Lambda_n$.
%In the case that 
Given $G = (V,E)$ and defining $\mathbb P$ to be the set of 
positive integers, we call a function $\kappa: V \rightarrow \mathbb P$
a {\em proper coloring} of $G$ if $\kappa(u) \neq \kappa(v)$ whenever 
$(u,v) \in E$.
%$\asc(\kappa) = \# \{(u,v) \in E \,|\, u<v \text{ and } \kappa(u)<\kappa(v) \}$.
%\end{equation*}
Then we have the definition
\begin{equation*}
%\label{eq:xpqdef}
X_G = \sum_\kappa x_{\kappa(1)} \cdots x_{\kappa(n)},
\end{equation*}
where the sum is over all proper colorings of $G$.
When $G$ is the incomparability graph of a poset $P$, we will
write $X_P = X_{\text{inc}(P)}$.
Stanley showed~\cite[Prop.\,2.4]{StanSymm} that in this case, 
we have the equivalent definition
%have the equivalent definition of
%$\{ X_P \,|\, P \text{ an {\em n}-element poset}\,\}$
%by 
\begin{equation}\label{eq:cPdef}
X_P = \sum_{\lambda \vdash n} c_{P,\lambda} m_\lambda,
\end{equation}
where $c_{P,\lambda}$ is the number of ordered set partitions of $P$ 
whose blocks are 
chains of cardinalities $\lambda_1, \lambda_2, \dotsc$.
%(More generally, chromatic symmetric functions were defined for arbitrary
%graphs. 
%(See \cite[Prop.\,2.4]{StanSymm}.)
These symmetric functions are related to $\sn$-class function 
evaluations as follows.
\begin{thm}\label{t:chromeq}
Let $P$ be an $n$-element unit interval order, let $v \in \sn$ 
be the 
%corresponding 
%\pavoiding 
$312$-avoiding permutation satisfying $\zeta(F_v) = P$
as in the proof of Theorem~\ref{t:dsnuio},
and let $w \in \sn$ be any \pavoiding permutation satisfying $w \sim v$ 
as in (\ref{eq:posetequiv}).
%.  
Then we have 
%$X_{C'_w(1)} = X_{P(F_w)}$.
\begin{equation}\label{eq:chromeq}
X_P = \sum_{\lambda \vdash n} \epsilon^\lambda(C'_w(1))m_\lambda.
\end{equation}
\end{thm}
\begin{proof}
It is easy to see that $c_{P,\lambda}$ is equal to the number of column-strict
$P$-tableaux of shape $\lambda^\tr$.  
By Theorems~\ref{t:dsnuio},
%this is 
%the number of column-strict $F_v$-tableaux 
%of shape $\lambda^\tr$,
%and by Theorem~
\ref{t:thetavw}, this is
% it is also  
the number of column-strict $F_w$-tableaux of shape $\lambda^\tr$.
Thus by Theorem~\ref{t:sncfinterp} (i) we have
%it is 
$c_{P,\lambda} = \epsilon^\lambda(C'_w(1))$.
%This follows from
%%Let $v \in \sn$ be the unique $312$-avoiding permutation satisfying
%%$v \sim w$.  Then by 
%the definition (\ref{eq:cPdef}), 
%and Theorems~\ref{t:thetavw}, 
%%and
%%Theorem
%~\ref{t:sncfinterp}.
%%we have
%%\begin{equation}
%%X_{C'_w(1)} \sum_{\lambda \vdash n} \epsilon^\lambda(C'_v(1))m_\lambda
\end{proof}
Expanding $X_P$ in other bases of $\mathbb Q \otimes \Lambda_n$, 
including the forgotten basis $\{ f_\lambda \,|\, \lambda \vdash n \}$, 
we see that other class function evaluations appear as coefficients.
%have the following.
\begin{cor}\label{c:chromeq}
Let $P$, $v$, $w$ be as in Theorem~\ref{t:chromeq}.
Then we have
\begin{equation*}
%\label{eq:chromeq}
\begin{aligned}
X_P &= 
%X_P = 
\sum_{\lambda \vdash n} 
\eta^{\lambda}(C'_w(1)) f_\lambda
= \sum_{\lambda \vdash n}  
\chi^{\lambda^\tr} (C'_w(1)) s_\lambda\\
&= \sum_{\lambda \vdash n}  
%= \sum_{\lambda \vdash n}  
\frac{(-1)^{n - \ell(\lambda)}}{z_\lambda} \psi^{\lambda}(C'_w(1)) p_\lambda
%= \ntnsp \sum_{\lambda \vdash n} \ntnsp 
= \sum_{\lambda \vdash n}  
\phi^{\lambda}(C'_w(1)) e_\lambda.
\end{aligned}
\end{equation*}
\end{cor}
\begin{proof}
The transition matrices relating the class function bases
$\{\epsilon^\lambda \,|\, \lambda \vdash n \}$,
$\{\eta^\lambda \,|\, \lambda \vdash n \}$, 
$\{\chi^{\lambda^\tr} \,|\, \lambda \vdash n \}$, 
$\{(-1)^{n-\ell(\lambda)}z_\lambda^{-1} \psi^\lambda \,|\, \lambda \vdash n \}$, 
$\{\phi^\lambda \,|\, \lambda \vdash n \}$, 
respectively,
are inverse to those relating the symmetric function bases
$\{m_\lambda \,|\, \lambda \vdash n \}$,
$\{f_\lambda \,|\, \lambda \vdash n \}$,
$\{s_\lambda \,|\, \lambda \vdash n \}$,
$\{p_\lambda \,|\, \lambda \vdash n \}$,
$\{e_\lambda \,|\, \lambda \vdash n \}$,
respectively.
\end{proof}

We remark that Theorem~\ref{t:chromeq} and Corollary~\ref{c:chromeq}
do not hold for arbitrary $w$ and $P$.  
Not all chromatic symmetric functions $X_P$ can
be expressed 
as $\sum_{\lambda \vdash n} \epsilon^\lambda(C'_w(1)) m_\lambda$
for appropriate $w \in \sn$,
nor can all 
symmetric functions of this form
be expressed as $X_P$
for an appropriate labeled poset $P$.

%Motivated by Theorem~\ref{t:chromeq}, we define 
%for each element $g \in \zsn$ a symmetric function 
%%$X_g \in \mathbb C \otimes \Lambda_n$
%$X_g \in \Lambda_n$
%by
%%\begin{equation}\label{eq:xgdef}
%$X_g = \sum_{\lambda \vdash n} \epsilon^\lambda(g) m_\lambda$.
%%\end{equation}
%In particular, we have the element
%%\begin{equation}\label{eq:chromdef}
%%$X_{C'_w(1)} = \sum_{\lambda \vdash n} \epsilon^\lambda(C'_w(1)) m_\lambda$,
%$X_{C'_w(1)}$
%% = \sum_{\lambda \vdash n} \epsilon^\lambda(C'_w(1)) m_\lambda$,
%%\end{equation}
%which by Theorems~\ref{t:thetavw} and \ref{t:chromeq} 
%is equal to $X_{P(F_w)}$ when $w$ \avoidsp.
%Thus $X_{C'_w(1)}$ is a generating function for values
%$\{ \theta^\lambda(C'_w(1)) \,|\, \lambda \vdash n \}$,
%where $\{ \theta^\lambda \,|\, \lambda \vdash n \}$ 
%is any one of the common bases of the space of
%$\sn$-class functions.
%On the other hand, not all chromatic symmetric functions $X_P$ can
%be expressed as $X_{C'_w(1)}$ for appropriate $w \in \sn$,
%nor can all 
%generating functions $X_{C'_w(1)}$ be expressed as $X_P$
%for an appropriate poset $P$.

Stanley and Stembridge~\cite[Conj.\,5.1]{StanSymm}, 
\cite[Conj.\,5.5]{StanStemIJT}
conjectured that $X_P$ is elementary nonnegative 
when $P$ is
$(\mathbf{3}+\mathbf{1})$-free, and
Gasharov~\cite[Thm.\,2]{GashInc} proved 
the weaker statement that $X_P$ is Schur nonnegative in this case.
% for such posets.
Guay-Paquet~\cite[Thm.\,5.1]{GPModRel} showed that the above conjecture
and result are equivalent to the analogous statements in which $P$ is assumed
to be a unit interval order.
Thus by Theorem~\ref{t:thetavw} and Corollary~\ref{c:chromeq}
%\ref{t:chromeq} 
%and the fact that
%unit interval orders are a proper subset of 
%$(\mathbf{3}+\mathbf{1})$-free posets, 
the nonnegativity statements
%conjecture and result 
are consequences
%special cases
%the elementary nonnegativity
%and Schur nonnegativity of the symmetric functions
%$\sum_{\lambda \vdash n}\epsilon^\lambda(C'_w(1))m_\lambda$
%for $w$ \avoidingp.
%Elementary nonnegativity of these symmetric functions
%for all $w$ is equivalent to
%the 
%of the $q=1$ specialization 
of Haiman's conjecture and result that 
for all $w \in \sn$, $\lambda \vdash n$
we have
$\phi_q^\lambda(\qew C'_w(q)) \in \mathbb N[q]$
%~\cite[Conj.\,2.1]{HaimanHecke} 
and
%$\phi_q^\lambda(\qew C'_w(q)), \chi_q^\lambda(\qew C'_w(q)) 
$\chi_q^\lambda(\qew C'_w(q)) \in \mathbb N[q]$, 
respectively \cite[Conj.\,2.1, Lem.\,1.1]{HaimanHecke}.
%for all $w \in \sn$, $\lambda \vdash n$
Specifically, the nonnegativity statements are obtained from Haiman's by 
restricting to the case that
$w$ 
avoids the pattern $312$
%\avoidsp\ 
%(or simply avoids the pattern $312$)
and by specializing at $q=1$.
% permutations
%statements at $q=1$ and assume that $w$ \avoidsp.


%The term {\em chromatic symmetric function}
%was coined by Stanley and Stembridge~\cite{StanSymm}, \cite{StanStemIJT},
%who associated a symmetric function $X_P$ to each poset $P$.
% associated to $P$} 
% symmetric function
%It is known that 
%Our use of the term {\em chromatic} comes from the fact that 
%Specifically, we have the following.

%($\star$ Mention somewhere that for
%$w \in \sn$ \avoidingp, there exists a $312$-avoiding permutation $v$
%such that we have $X_{C'_w(1)} = X_v$.)
%\end{thm}
%\begin{proof}
%Omitted.  Construct $v$ so that $P(F_v) = P(F_w)$.
%\end{proof}

%if $w$ avoids the pattern $312$, then 
%our symmetric function 
%$X_{C'_w(1)}$
%is equal to the Stanley-Stembridge chromatic symmetric function $X_{P(F_w)}$.
%definition is consistent
%there is a unit interval order ($\star$ define?) $P = P(w)$
%such that our definition is consistent with the original definition of $X_P$,
%the {\em chromatic symmetric function associated to $P$} 
%for which $X_{C'_w(1)}$ is equal to the symmetric function $X_P$ defined 
%in \cite{StanSymm}, \cite{StanStemIJT}.
%, \cite{StanSymm}.
%On the other hand, it is not true that for all posets $P$, 
%we may express $X_P$ as $X_{C'_w(1)}$ for an appropriate permutation $w$.  
%Neither is it true that for all permutations
%$w$, we may express $X_{C'_w(1)}$ as $X_P$ for an appropriate poset $P$.
%($\star$ Define $\lambda^\tr$ earlier?)
%= \sum_{\lambda \vdash n} \imm{\phi^\lambda}(A) e_\lambda,
%\end{aligned}
%\end{equation}
%A well-known $f$-nonnegativity problem
%special case of Problem~\ref{p:fnonneg} 
%concerns a certain
%{\em chromatic symmetric function} $X_P$ associated to each labeled poset $P$.
%In particular, 

%($\star$ Rewrite the following to explicitly credit Shareshian-Wachs for 
%the definition which inspired Equation~\ref{eq:qchromdef}?)
%is an elementary
%nonnegative symmetric function.
%\begin{thm}
%Let $w$ avoid the patterns $3412$, $4231$.  Then $X_{C'_w(1)}$ is an elementary
%nonnegative symmetric function.
%\end{thm}
%\begin{proof}
%Omitted.
%\end{proof}

Let $\mathrm{QSym}_n$ be the $\mathbb Z$-module 
of homogeneous degree $n$ {\em quasisymmetric functions}
in the commuting indeterminates $x_1, x_2, \dotsc$,
%i.e., the bounded degree power series 
%in which all monomials of the forms 
i.e., the generalization of homogeneous degree $n$ symmetric functions 
$\Lambda_n$ in which monomials of the forms
%defined by the requirement that all monomials of the forms
%having the property
%satisfying the requirement 
%that all monomials of the forms
%bounded degree power series in which all monomials of the forms 
%of the forms
$x_1^{\alpha_1} \cdots x_k^{\alpha_k}$ and
$x_{i_1}^{\alpha_1} \cdots x_{i_k}^{\alpha_k}$ 
are required to 
have the same
coefficient only when $i_1 < \cdots < i_k$.
%(See \cite[Ch.\,7]{StanEC2} for more information.)
% on quasisymmetric functions.)
In \cite[Sec.\,4]{SWachsChromQ}, 
%In \cite[Defn.\,1.2]{SWachsChromQF}, 
%Shareshian and Wachs defined a certain $q$-analog
Shareshian and Wachs defined a $q$-analog
$X_{G,q}$ 
of the chromatic symmetric function $X_G = X_{G,1}$, with
%we have
$\{ X_{G,q} \,|\, G \text{ a simple labeled graph on $n$ vertices}\,\}$ 
belonging to $\mathbb Z[q] \otimes \mathrm{QSym}_n$,
and studied expansions of these functions in various bases.
%Following \cite[Defn.\,1.2]{SWachsChromQF}, 
Assume
that $V$ is labeled by $[n]$ and let
%\begin{equation*}
%Given a graph $G = (V,E)$ and defining $\mathbb P$ to be the set of 
%positive integers, we call a function $\kappa: V \rightarrow \mathbb P$
%a {\em proper coloring} of $G$ if $\kappa(u) \neq \kappa(v)$ whenever 
%$(u,v) \in E$.
$\asc(\kappa) = \# \{(u,v) \in E \,|\, u<v \text{ and } \kappa(u)<\kappa(v) \}$.
%\end{equation*}
Then we have the definition
\begin{equation*}
%\label{eq:xpqdef}
X_{G,q} = \sum_\kappa q^{\asc(\kappa)} x_{\kappa(1)} \cdots x_{\kappa(n)},
\end{equation*}
where the sum is over all proper colorings of $G$.
In \cite[Thm.\,4.5]{SWachsChromQF} 
Shareshian and Wachs showed 
that this function 
%$X_{G,q}$ 
%turns out to be
is in fact 
%is 
symmetric 
with coefficients in $\mathbb Z[q]$
%(belongs to the $\mathbb Z$-module $\mathbb Z[q] \otimes \Lambda_n$ of 
%homogeneous degree $n$ symmetric functions with coefficients in $\mathbb Z[q]$)
%under certain conditions.
%and studied expansions of these functions in various bases.
%$\mathbb Q[q] \otimes \mathrm{QSym}_n$.
%In the case that 
when
%When 
$G$ is the incomparability graph of an appropriately labeled
$n$-element unit interval order
%labeled poset 
$P$.  
Specifically, we require
%labeled so that
%for all 
for each pair $x,y \in P$ satisfying
%, the inequality
\begin{equation}\label{eq:altrespect}
\# \{ z \in P \,|\, z < x \} - 
\# \{ z \in P \,|\, z > x \}
<
\# \{ z \in P \,|\, z < y \} - 
\# \{ z \in P \,|\, z > y \},
\end{equation}
%implies that 
that the label of $x$ be less than that of $y$.
(The equivalence of this requirement to that stated in
\cite[Thm.\,4.5]{SWachsChromQF} 
follows from comparison of 
%sees this 
%if one compares 
\cite[Props.\,4.1 -- 4.2]{SWachsChromQF}
and the definition of 
{\em natural unit interval order}~\cite[Sec.\,4]{SWachsChromQF} to 
results in \cite[p.\,33]{Fish} and 
\cite[Obs.\,2.1 -- 2.3, Prop.\,2.4]{SkanReed}.)
%$\mathbb Q[q] \otimes \mathrm{QSym}_n$.
%In the case that 
When $G$ is the incomparability graph of a labeled poset $P$, 
we will write $X_{P,q} = X_{\text{inc}(P),q}$, and 
%In this case 
we may give an alternate definition of
$\{ X_{P,q} \,|\, P \text{ an {\em n}-element poset}\,\}$
which is analogous to (\ref{eq:cPdef}).
%(This definition was known to Shareshian and Wachs:  
(See also \cite[Eq.\,(6.2)]{SWachsChromQF}.)
%in terms of poset colorings and a statistic $\asc$.
\begin{prop}\label{p:ascinv}
Let $P$ be a unit interval order, labeled as in (\ref{eq:altrespect}).
Then we have
\begin{equation*}
%\label{eq:qcPdef1}
X_{P,q} = \sum_{\lambda \vdash n} c_{P,\lambda}(q) m_\lambda,
\end{equation*}
where 
\begin{equation*}
%\label{eq:qcPdef2}
c_{P,\lambda}(q) = \sum_U q^{\inv(U)}
\end{equation*}
and the sum is over column-strict $P$-tableaux of shape $\lambda^\tr$.
\end{prop}
\begin{proof}
Each proper coloring $\kappa$ of $\mathrm{inc}(P)$
may be viewed as an assignment of colors to elements of $P$ so that
each subset of elements having a given color forms a chain.
By \cite[Thm.\,4.5]{SWachsChromQF}, $X_{P,q}$ is symmetric.
Thus for $\lambda \vdash n$, the coefficient in $X_{P,q}$ of $m_\lambda$ 
is well-defined: it is the coefficient of 
$x_1^{\lambda_r} x_2^{\lambda_{r-1}} \cdots x_r^{\lambda_1}$,
i.e., the sum of $q^{\asc(\kappa)}$, over all colorings $\kappa$ 
that assign 
color $1$ to a $\lambda_r$-element chain,
color $2$ to a $\lambda_{r-1}$-element chain$,\dotsc,$
color $r$ to a $\lambda_1$-element chain.
Each such coloring corresponds to 
%, we may construct 
a column-strict 
$P$-tableau $U$ of shape $\lambda$.  Specifically, 
each $\lambda_{r+1-i}$-element chain of color $i$ 
corresponds to column $r+1-i$ of $U$, for $i = 1, \dotsc, r$.
Now observe that a pair $(u,v)$ in $P$, with $u < v$ as integers,
forms an ascent of $\kappa$ if and only if it forms an inversion in $U$.  
Specifically, $(u,v)$ is an edge in $\mathrm{inc}(P)$ 
if and only if $u$, $v$ are incomparable in $P$, and we have
$\kappa(u) < \kappa(v)$ if and only if $u$ appears in a column of $U$
to the right of the column containing $v$.
\end{proof}

%Thus the $q=1$ specialization of $X_{P,q}$ is just $X_P$.
Just as
Stanley's 
chromatic symmetric functions $X_P$ are related 
to $\sn$-class function evaluations in Theorem~\ref{t:chromeq},
%e will now show that 
the Shareshian-Wachs 
chromatic quasisymmeric functions $X_{P,q}$
are related to $\hnq$-trace evaluations.
%in the same way 
%that 
%These symmetric functions are related to $\hnq$-trace evaluations as follows.
\begin{thm}\label{t:qchromeq}
Let $P$ be an $n$-element unit interval order labeled as in 
(\ref{eq:altrespect}),
let $v \in \sn$ 
be the corresponding 
%\pavoiding 
$312$-avoiding permutation as in Theorem~\ref{t:dsnuio},
and let $w \in \sn$ be any \pavoiding permutation satisfying $w \sim v$ 
as in (\ref{eq:posetequiv}).
%.  
Then we have 
%$X_{C'_w(1)} = X_{P(F_w)}$.
\begin{equation*}
X_{P,q} = \sum_{\lambda \vdash n} \epsilon_q^\lambda(\qew C'_w(q))m_\lambda.
\end{equation*}
\end{thm}
\begin{proof}
By Proposition~\ref{p:ascinv}, $c_{P,\lambda}(q)$ 
is equal to the sum of $q^{\inv(U)}$ over column-strict
$P$-tableaux of shape $\lambda^\tr$.  
By Theorems~\ref{t:dsnuio}, \ref{t:qthetavw}, 
we may 
%this is equal to the same 
sum over column-strict 
%$F_v$-tableaux, 
%of shape $\lambda^\tr$,
%and by Theorem~\ref{t:qthetavw}, 
%it is also the the same 
%we may sum 
%over column-strict 
$F_w$-tableaux.
% of shape $\lambda^\tr$.
%Thus by 
Now Theorem~\ref{t:qepsilon} gives
%we have
%it is 
$c_{P,\lambda}(q) = \epsilon^\lambda(\qew C'_w(q))$.
%Comparing Theorem~\ref{t:qepsilon} and Equations 
%(\ref{eq:qcPdef1}) -- (\ref{eq:qcPdef2}), we see that 
%$c_{P,q}$ is equal to $\epsilon_q^\lambda(\qew C'_w(q))$.
%By Theorem~\ref{t:sncfinterp} (i), this is $\epsilon^\lambda(C'_w(1))$.
\end{proof}

Expanding $X_P$ in other bases of $\mathbb Q[q] \otimes \Lambda_n$
and following the proof of Corollary~\ref{c:chromeq}, 
%including the forgotten basis $\{ f_\lambda \,|\, \lambda \vdash n \}$, 
we see that other trace evaluations appear as coefficients.
%we have the following.
\begin{cor}\label{c:qchromeq}
Let $P$, $v$, $w$ be as in Theorem~\ref{t:qchromeq}.
Then we have
\begin{equation*}
\begin{aligned}
X_{P,q} &= 
%\sum_{\lambda \vdash n} \epsilon_q^\lambda(\qew C'_w)m_\lambda
%= 
\sum_{\lambda \vdash n} 
\eta_q^{\lambda}(\qew C'_w(q)) f_\lambda
%= \ntnsp \sum_{\lambda \vdash n} \ntnsp 
= \sum_{\lambda \vdash n}  
%\chi_q^{\lambda^\tr} \ntnsp (\qew C'_w) s_\lambda\\
\chi_q^{\lambda^\tr} (\qew C'_w(q)) s_\lambda\\
%= \sum_{\lambda \vdash n} \imm{\chi^{\lambda^\tr}}(A) s_\lambda\\
%= \sum_{\lambda \vdash n} \ntnsp \frac{\psi_q^{\lambda}(\qew C'_w)}{(-1)^{n - %ell(\lambda)}z_\lambda} p_\lambda
%&= \ntnsp \sum_{\lambda \vdash n} \ntnsp 
&= \sum_{\lambda \vdash n}  
%\frac{\psi_q^{\lambda}(\qew C'_w)}{(-1)^{n - \ell(\lambda)}} \frac{p_\lambda}{%_\lambda}
\frac{(-1)^{n - \ell(\lambda)}}{z_\lambda} \psi_q^{\lambda}(\qew C'_w(q)) p_\lambda
%= \sum_{\lambda \vdash n} (-1)^{n - \ell(\lambda)}\frac{\psi_q^{\lambda}(\qew %C'_w)}{z_\lambda} p_\lambda\\
%= \ntnsp \sum_{\lambda \vdash n} \ntnsp 
= \sum_{\lambda \vdash n}  
\phi_q^{\lambda}(\qew C'_w(q)) e_\lambda .
%\end{gathered}
\end{aligned}
\end{equation*}
%\end{equation*}
\end{cor}
%\begin{proof}
%See proof of Corollary~\ref{c:chromeq}.
%\end{proof}

%Like Theorem~\ref{t:chromeq} and Corollary~\ref{c:chromeq},
As before,
Theorem~\ref{t:qchromeq} and Corollary~\ref{c:qchromeq}
do not hold for arbitrary $w$ and $P$.  
%, we define for each element $g \in \hnq$ a
%symmetric function $X_g \in \zqq \otimes \Lambda_n$
%by $X_g = \sum_{\lambda \vdash n} \epsilon_q^\lambda(g) m_\lambda$.
%In particular, we have the element
%\begin{equation}\label{eq:qchromdef}
%X_{\qew C'_w(q)} = \sum_{\lambda \vdash n} \epsilon_q^\lambda(\qew C'_w(q)) m_\%lambda
%\end{equation}
%which by Theorems~\ref{t:thetavw} and \ref{t:qchromeq} 
%is equal to $X_{P(F_w),q}$ when $w$ \avoidsp.
%Clearly
%the function $X_{\qew C'_w(q)}$ specializes at $q=1$ to $X_{C'_w(1)}$.
%Again, not all 
Not all chromatic symmetric functions $X_{P,q}$ can
be expressed 
as $\sum_{\lambda \vdash n} \epsilon_q^\lambda(\qew C'_w(q)) m_\lambda$
%$X_{\qew C'_w(q)}$ 
for appropriate $w \in \sn$,
nor can all 
%quantum symmetric trace generating 
symmetric functions of this form
%$X_{\qew C'_w(q)}$ 
be expressed as $X_{P,q}$
%$w$, we may express $X_{C'_w(1)}$ as $X_P$ 
for an appropriate labeled poset $P$.


Shareshian and Wachs conjectured
%\cite[Conj.\,7.6]{SWachsChromQF} 
%conjectured 
that 
%we have 
$X_{P,q}$ belongs to 
%$X_{P,q} \in \spn_{\mathbb N[q]}\{e_\lambda \,|\, \lambda \vdash n \}$
$\spn_{\mathbb N[q]}\{e_\lambda \,|\, \lambda \vdash n \}$
% $\mathbb N[q]$elementary nonnegative 
%($\star$ define?)
when $P$ is a unit interval order labeled 
%$(\mathbf{3}+\mathbf{1})$-free,
%$(\mathbf{2}+\mathbf{2})$-free,
%and labeled appropriately, i.e., 
as in (\ref{eq:altrespect})~\cite[Conj.\,4.9]{SWachsChromQ}, 
and proved
%They proved the weaker statement~\cite[Thm.\,6.3]{SWachsChromQF}
that 
%we have
$X_{P,q}$ belongs to
%$X_{P,q} \in \spn_{\mathbb N[q]}\{s_\lambda \,|\, \lambda \vdash n \}$
$\spn_{\mathbb N[q]}\{s_\lambda \,|\, \lambda \vdash n \}$
for such posets~\cite[Thm.\,6.3]{SWachsChromQF}.
These statements do not hold for
the more general
%in greater generality for 
%the more general class of 
$(\mathbf 3 + \mathbf 1)$-free posets, since the functions $X_{P,q}$ are
not always symmetric in this case.
%for these posets.
Nevertheless,  
the result of Guay-Paquet~\cite[Thm.\,5.1]{GPModRel} shows that 
the statements generalize those of Stanley, Stembridge, and Gasharov
mentioned after Corrolary~\ref{c:chromeq}.
By Theorem~\ref{t:thetavw} and Corollary~\ref{c:qchromeq}, the statements
are special cases (corresponding to $w$ avoiding the pattern $312$)
%consequences 
of Haiman's conjecture and result~\cite[Conj.\,2.1, Lem\,1.1]{HaimanHecke}  
that for all $w \in \sn$, $\lambda \vdash n$
we have
%$\phi_q^\lambda(\qew C'_w(q)) \in \mathbb N[q]$~\cite[Conj.\,2.1]{HaimanHecke} 
$\phi_q^\lambda(\qew C'_w(q)) \in \mathbb N[q]$
%~\cite[Conj.\,2.1]{HaimanHecke} 
and
$\chi_q^\lambda(\qew C'_w(q)) \in \mathbb N[q]$.
%~\cite[Lem.\,1.1]{HaimanHecke}.
%that we have $\phi_q^\lambda(\qew C'_w(q)) \in \mathbb N[q]$ for all 
%$w \in \sn$, $\lambda \vdash n$~\cite[Conj.\,2.1]{HaimanHecke}
%is equivalent to the containment
%of $X_{\qew C'_w(q)}$ in $\spn_{\mathbb N[q]}\{e_\lambda \,|\, \lambda \vdash n \}$ 
%for all $w \in \sn$.
%Similarly, his result 
%that we have $\chi_q^\lambda(\qew C'_w(q)) \in \mathbb N[q]$ for all 
%$w \in \sn$, $\lambda \vdash n$~\cite[Lem.\,1.1]{HaimanHecke}
%is equivalent to the containment
%of $X_{\qew C'_w(q)}$ in $\spn_{\mathbb N[q]}\{s_\lambda \,|\, \lambda \vdash n \}$ 
%for all $w \in \sn$.
%Combining Theorem~\ref{t:thetavw} with the 
%Shareshian-Wachs elementary nonnegativity
%conjecture and Schur nonnegativity result, we obtain
%the special cases of Haiman's conjecture and lemma
%corresponding to $w$ \avoidingp.
Shareshian and Wachs also conjectured~\cite[Sec.\,7]{SWachsChromQF},
%~\cite[Conj.\,4.15--4.16]{SWachsChromQ}
and Athanasiadis proved~\cite[Thm.\,4]{AthanPSE}
that 
$X_{P,q}$ belongs to 
$\spn_{\mathbb N[q]}
\{(-1)^{n - \ell(\lambda)} z_\lambda^{-1} p_\lambda \,|\, \lambda \vdash n \}$
when $P$ is a unit interval order labeled as in (\ref{eq:altrespect}).
%we have that 
By Theorem~\ref{t:thetavw} and Corollary~\ref{c:qchromeq}
this is equivalent to the assertion that
%$\psi_q^\lambda(\qew C'_w(q))$ belongs to $\mathbb N[q]$ for $w$ avoiding
we have $\psi_q^\lambda(\qew C'_w(q)) \in \mathbb N[q]$ for $w$ avoiding
the pattern $312$.  
Thus this result is a special case of the (unpublished) conjecture
that $\psi_q^\lambda(\qew C'_w(q)) \in \mathbb N[q]$ for all $w \in \sn$,
$\lambda \vdash n$, 
which is a weakening of Haiman's conjecture~\cite[Conj.\,2.1]{HaimanHecke}
since $\psi_q^\lambda$ is a nonnegative linear combination of monomial traces
(\ref{eq:moncfdef}).

%(See Theorems~\ref{t:swqpsi}, \ref{t:qpsiO}, \ref{t:qpsi} for combinatorial
%interpretations of $\psi_q^\lambda(\qew C'_w(q))$.)
%belongs to
%$\spn_{\mathbb N[q]} \{\phi_q^\lambda \,|\, \lambda \vdash n \}$,
%this provides the first proof of a special case 
%of a weakening of 
%the unstated conjecture that
%we have
%$\psi_q^\lambda(\qew C'_w(q))$ belongs to $\mathbb N[q]$ for all $w \in \sn$.  
%Section~\ref{s:hnqinterpp}.)

%$\star$ Mention that Shareshian and Wachs have interpretations
%of the coefficient of $q^k$ in $X_{P,q}$, and state the problem
%of interpreting the same coefficient in $X_{\qew C'_w(q)}$.

\section{Interpretation of $\chi_q^\lambda(\qew C'_w(q))$}\label{s:hnqinterps}
Combining results in Sections~\ref{s:pathposet}, \ref{s:hnqinterph}, 
\ref{s:chromsf}
with those of
%Corollary~\ref{c:qchromeq} with results of 
Shareshian and Wachs
now leads to the following $q$-analog of Theorem~\ref{t:sncfinterp} (iii).

\begin{thm}\label{t:qchi}
Let $w \in \sn$ \avoidp.
% the pattern $312$.
%patterns $3412$, $4231$.  
For $\lambda \vdash n$ we have
%\begin{equation}\label{eq:epqinterp}
%\epsilon_q^{\lambda}(\qew C'_w(q)) = \sum q^{\inv(U)},
%\end{equation}
%where the sum is over all column-strict $F_w$-tableaux $U$ of type $e$ and
%shape $\lambda^\tr$.
%We also have
\begin{equation}\label{eq:chiqinterp}
\chi_q^\lambda(\qew C'_w(q)) = \sum_U q^{\inv(U)},
\end{equation}
where the sum is over all standard $F_w$-tableaux 
%$U$ 
of type $e$ and shape $\lambda$.
\end{thm}
\begin{proof}
Let $P = P(F_w)$.
%Let $v$ be the unique $312$-avoiding permutation satisfying $v \sim w$
%(\ref{eq:posetequiv}),
%and let $P$ be the unit interval order corresponding to $v$ as in 
%Theorem~\ref{t:dsnuio}.
%By Proposition~\ref{p:}, we have
%$\chi_q^\lambda(\qew C'_w(q)) = \chi_q^\lambda(\qev C'_v(q))$,
%and by 
By Corollary~\ref{c:qchromeq}, $\chi_q^\lambda(\qew C'_w(q))$ is equal 
to the coefficient of $s_{\lambda^\tr}$ in $X_{P,q}$.
By \cite[Thm.\,6.3]{SWachsChromQF}
and Theorems~\ref{t:dsnuio}, \ref{t:qthetavw}, 
this is precisely the claimed sum.
\end{proof}

%Let $\Sigma(1,w,\lambda)$ be the sum on the right-hand side above,
%and let $\Sigma(2,w,\lambda)$ be the sum on the right-hand side of 
%(\ref{eq:epsilonmain}).
%A result of Shareshian and Wachs~\cite{SWachsChromQ} implies that the two sets 
%$\{ \Sigma(1,w,\lambda) \,|\, \lambda \vdash n \}$ and
%$\{ \Sigma(2,w,\lambda) \,|\, \lambda \vdash n \}$ are related by
%\begin{equation*}
%\Sigma(2,w,\lambda) = \sum_{\mu} K_{\mu^\tr, \lambda} \Sigma(1,w,\mu),
%\end{equation*}
%where $(K_{\mu^\tr,\lambda})_{\lambda,\mu}$ is the 
%(invertible) matrix of Kostka numbers.
%Since we have
%\begin{equation*}
%\epsilon_q^\lambda(\qew C'_w(q)) = 
%\sum_{\mu} K_{\mu^\tr, \lambda} \chi_q^\mu(\qew C'_w(q)),
%\end{equation*}
%the result follows from Theorem~\ref{t:qepsilon}.
%\end{proof}

For example, consider again the descending star network $F_{3421}$ in
(\ref{eq:F3421}).
It is easy to verify that there are exactly two standard
$F_{3421}$-tableaux of type $e$ and shape $31$:
the two column-strict $F_{3421}$-tableaux of type $e$ and shape $31$
in (\ref{eq:epsilontabs31}) are also row-semistrict.
Thus we have
$\chi_q^{31}(q_{e,3421} C'_{3421}(q)) 
= \epsilon_q^{211}(q_{e,3421} C'_{3421}(q)) = q^2 + q^3$.
On the other hand, not all twenty-four $F_{3421}$-tableaux of type $e$ and 
shape $4$ are row-semistrict: the six tableaux with $\pi_4$ immediately
preceding $\pi_1$ are not.
It is easy to verify that the eighteen remaining tableaux
give $\chi_q^{4}(q_{e,3421} C'_{3421}(q)) = 1 + 3q + 5q^2 + 5q^3 + 3q^4 + q^5$.
Since there are no column-strict $F_{3421}$-tableaux of shapes $22$, $211$, or
$1111$, there are no standard $F_{3421}$-tableaux of these shapes either,
and we have
$\chi_q^{22}(q_{e,3421} C'_{3421}(q)) = 
\chi_q^{211}(q_{e,3421} C'_{3421}(q)) = 
\chi_q^{1111}(q_{e,3421} C'_{3421}(q)) = 0$.

Combining Theorems~\ref{t:kstar} and \ref{t:qchi}, we have the following
analog of Corollary~\ref{c:edeck}. 
\begin{cor}\label{c:sdeck}
Fix $\lambda = (\lambda_1, \dotsc, \lambda_r) \vdash n$
and $w \in \sn$ \avoidingp.  If $w_1 \cdots w_n$ has 
a decreasing subsequence of length greater than $\lambda_1$, then
we have $\chi_q^\lambda(\qew C'_w(q)) = 0$.
\end{cor}

More generally, 
it is known that we have $\chi_q^\lambda(\qew C'_w(q)) = 0$
unless $\lambda \geq \sh(w)^\tr$ in the majorization order.
(See the comment following Corollary~\ref{c:edeck}.)
%, where
%$\sh(w)$ is the partition associated to $w$ 
%by the Robinson-Schensted row insertion algorithm. 
%(See, e.g., \cite[Prop.\,4.1\,(3)]{HaimanHecke}.)
This implies Corollary~\ref{c:sdeck}, since a decreasing subsequence
of length greater than $\lambda_1$
%$\lambda_1^{\tr}$ 
in $w_1 \dotsc w_n$ implies that
we have $\lambda \ngeq \sh(w)^\tr$.


%($\star$ Explain.)
%Let $U$ be an $F$-tableau of shape $\lambda = (\lambda_1, \dotsc, \lambda_r)$
%containing a path family $\pi$, 
%and let $U_i$ be the $i$th row of $U$.
%Let $U_1 \circ \cdots \circ U_r$ and $U_r \circ \cdots \circ U_1$
%be the $F$-tableaux of shape $n$ consisting of the rows of $U$ concatenated
%in increasing and decreasing order, respectively.
% and let 
%$U_{i,j}$ 
%$\pi_{i,1}, \dotsc, \pi_{i,\lambda_i}$ 
%denote the path in row $i$, position $j$ of $U$. 
%Define the {\em flattened $F$-tableau} $\ol{U}$ corresponding to $U$ to be 
%the tableau of shape $n$ which is the concatenation of rows $1,\dotsc,r$,
%\begin{equation}\label{eq:flattab}
%%\ol U = (\pi_{1,1}, \dotsc, \pi_{1,\lambda_1}, \pi_{2,1}, \dotsc, \pi_{r,\lambda_r}).
%\ol U = (U_{1,1}, \dotsc, U_{1,\lambda_1}, 
%U_{2,1}, \dotsc, U_{2,\lambda_2}, 
%\dotsc,
%U_{r,1}, \dotsc, U_{r,\lambda_r}).
%\end{equatio


%\section{Conjectured interpretations of $\hnq$-traces}\label{s:chnqinterp}
\section{Interpretation of $\psi_q^\lambda(\qew C'_w(q))$}\label{s:hnqinterpp}

%For $\lambda \vdash n$ and
%$w \in \sn$ \avoidingp,
%Theorem~\ref{t:sncfinterp} (iv-a) -- (v-b) interprets 
%$\psi^\lambda(C'_w(1))$ and
%$\phi^\lambda(C'_w(1))$ (for special cases of $\lambda$)
%as cardinalities of certain sets of $F_w$-tableaux.
%There are no known or even conjectured interpretations of
%$\phi^\lambda(C'_w(1))$ for general $\lambda$.
%%\ssec{Interpretations of evaluations of $\hnq$-traces}
%While there are no known $q$-analogs of 
%Theorem~\ref{t:sncfinterp} (iv-a) -- (v-b),
%we will use these same sets of $F_w$-tableaux and 
%variations of the statistic $\inv$ on $P$-tableaux,
%to conjecture
%that 
%$\psi_q^\lambda(\qew C'_w(q))$, and
%$\phi_q^\lambda(\qew C'_w(q))$ (for special $\lambda$)
%are generating functions for
%tableaux on which the statistics take the values $k = 0,1,\dotsc$.

%$\star$ Say something about $\psi_q^\lambda$ being understood for $q=1$, but
%that there are no results for general $q$.
%$\star$ Say something about $\phi_q^\lambda$ not being understood even
%for $q=1$, except in special cases, and that there are no results for 
%general $q$.

%\section{Interpretation of $\psi_q^\lambda(\qew C'_w(q))$}
%\subsection{Interpretation of $\psi_q^\lambda(\qew C'_w(q))$}

%($\star$ Check the results in this subsection
%to ensure that they hold for \pavoiding permutations.)

%($\star$ Include special cases that Shareshian and Wachs have already proved.)

%($\star$ Mention which parts of Theorem 4.7 correspond to these conjectures.)

%Shareshian and Wachs conjectured and proved the equivalence of
%two formulas for
%coefficients in the power sum expansion of 
%$X_{P,q}$~\cite[Conj.\,7.6, Lem.\,7.7, Prop.\,7.8]{SWachsChromQF}. 
%Athanasiadis proved these formulas in \cite[Thm.\,3.1]{AthanPSE}.
%Defining $[a]_q = 1 + q + \cdots + q^{a-1}$
%for $a \geq 1$ and $[0]_q = 0$,
%and applying
%Corollary~\ref{c:qchromeq} to the above results, 
%we obtain the following formulas for
%$\psi_q^\lambda(\qew C'_w(q))$,
%i.e., $q$-analogs of Theorem~\ref{t:sncfinterp} (iv-b)--(iv-d).
Combining results in Sections~\ref{s:pathposet}, \ref{s:hnqinterph}, 
\ref{s:chromsf}
with those of
%Corollary~\ref{c:qchromeq} with results of 
Shareshian, Wachs, and Athanasiadis
% and Athanasiadis
now leads to 
%the following 
%formulas for
%can state these results 
%can be stated 
%in terms of 
%the $\hnq$ traces $\psi_q^\lambda$:
%$\psi_q^\lambda(\qew C'_w(q))$,
%i.e., 
$q$-analogs of Theorem~\ref{t:sncfinterp} (iv-c)--(iv-d).
We will sometimes find it useful to 
reflect path tableaux in a vertical line,
and will write $U^R$ for the {\em reverse} of tableau $U$.
For instance, we have
\begin{equation*}
U = \raisebox{-6mm}{\includegraphics[height=12mm]{tableaux/31_1234}}\ , \qquad
U^R = \raisebox{-6mm}{\includegraphics[height=12mm]{tableaux/31_1234r}}\ ,
\end{equation*}
where each path $\pi_i$ retains its original source, sink, and orientation.
Thus $U_i^R$ will denote the reverse of the $i$th row of $U$.
Note that while $U^R$ may not be a tableau, 
because its cells are right-justified rather than left justified,
the functions $\inv$ and $\rinv$ may still be applied to $U^R$ as
at the end of Section~\ref{s:pathposet}.
We will also use standard notation for the $q$-analogs of the nonnegative
integers and factorial function.  For $a \in \mathbb N$ we define
%Define 
$[a]_q = 1 + q + \cdots + q^{a-1}$
for $a \geq 1$, and $[0]_q = 0$.
We also define
$[a]_q! = [a]_q [a-1]_q \cdots [1]_q$ for $a \geq 1$, and $[0]_q! = 1$.
%\begin{equation*}
%[a]_q \defeq \begin{cases}
%1 + q + \cdots + q^{a-1} &\text{if $a \geq 1$},\\
%0 &\text{if $a = 0$},
%\end{cases}
%\end{equation*}

\begin{thm}\label{t:swqpsi}
Let $w \in \sn$ \avoidp. 
%and
%avoid the 
%pattern $312$.  
%patterns $3412$, $4231$.  
For 
$\lambda = (\lambda_1, \dotsc, \lambda_r) \vdash n$, we have
%\begin{equation}\label{etaqinterp}
%\eta_q^\lambda(\qew C'_w(q)) = \sum q^{\rinv(U_1 \circ \cdots \circ U_r)},
%\end{equation}
%where the sum is over all row-closed $F_w$-tableaux $U$ of shape $\lambda$.
%We also have
\begin{equation}\label{eq:swqpsi1}
%\smash{\psi_q^\lambda(\qew C'_w(q)) = \sum_U q^{\inv(U_r \circ \cdots \circ U_1)},}
\psi_q^\lambda(\qew C'_w(q)) = \sum_U q^{\inv(U_1 \circ \cdots \circ U_r)},
\end{equation}
where the sum is over all record-free, row-semistrict
%cylindrical
%cyclically row-semistrict 
$F_w$-tableaux 
%$U$ 
of type $e$ and shape $\lambda$, and
%Equivalently, 
%Also, we have
\begin{equation}\label{eq:swqpsi2}
%\smash{\psi_q^\lambda(\qew C'_w(q)) = \sum_U q^{\inv(U_r \circ \cdots \circ U_1)},}
\psi_q^\lambda(\qew C'_w(q)) = [\lambda_1]_q \cdots [\lambda_r]_q
\sum_U q^{\inv(U_1^R \circ \cdots \circ U_r^R)},
\end{equation}
where the sum is over all right-anchored, row-semistrict
%cylindrical
%cyclically row-semistrict 
$F_w$-tableaux 
%$U$ 
of type $e$ and shape $\lambda$, 
%of shape $\lambda$ and type $e$.
%Shareshian-Wachs first and second conjecture, stated in our terms.
%Shareshian-Wachs first conjecture, stated in our terms.
\end{thm}
%\begin{conj}
%Shareshian-Wachs second conjecture, stated in our terms.
%\end{conj}
%Shareshian and Wachs proved the equivalence of 
%(\ref{eq:swqpsi1}) and (\ref{eq:swqpsi2}) 
%%the two parts of Conjecture~\ref{conj:swqpsi} 
%in ($\star$ where?)
\begin{proof}
Let $P = P(F_w)$.  By Corollary~\ref{c:qchromeq}, 
$\psi_q^\lambda(\qew C'_w(q))$ is equal to 
$(-1)^{n-r}z_\lambda$ times 
the coefficient of $p_\lambda$ in $X_{P,q}$.
By \cite[Thm.\,3.1]{AthanPSE} and Theorems~\ref{t:dsnuio}, \ref{t:qthetavw}, 
this is equal to the right-hand side of 
(\ref{eq:swqpsi1}).
By \cite[Lem.\,7.7]{SWachsChromQF} and 
Theorems~\ref{t:dsnuio}, \ref{t:qthetavw}, 
it is also equal to the right-hand side of
(\ref{eq:swqpsi2}). 
%Let $v$ be the unique $312$-avoiding permutation in $\sn$ which satisfies
%$v \sim w$.  
%Since all $F_w$-tableaux $U$ appearing in the sums in
%(\ref{eq:swqpsi1}) and (\ref{eq:swqpsi2}) 
%have type $e$, we may use
%Theorem~\ref{t:qthetavw} to replace these by
%$F_v$-tableaux (or $P(F_v)$-tableaux) 
%having the same properties.
%By Theorem~\ref{t:dsnuio}, the poset 
%$P = P(F_v)$ is a unit interval order, and
%by \cite[Lem.\,7.7]{SWachsChromQF} the right-hand sides of
%(\ref{eq:swqpsi1}) and (\ref{eq:swqpsi2}) 
%are therefore equal.
%By \cite[Thm.\,3.1]{AthanPSE}, the right-hand side of 
%(\ref{eq:swqpsi1}) is $(-1)^{n-r}z_\lambda$ times 
%the coefficient of $p_\lambda$ in $X_{P,q}$, as 
%had been 
%conjectured in
%\cite[Conj.\,7.6]{SWachsChromQF}.  By Corollary~\ref{c:qchromeq},
%this is $\psi_q^\lambda(\qew C'_w(q))$.
\end{proof}

%$\star$ Give examples?
For example, consider the descending star network 
$F_{3421}$ in (\ref{eq:F3421}) 
%and the first sum in 
and the sum in (\ref{eq:swqpsi1}).
It is easy to verify that there are 
eighteen record-free, row-semistrict $F_{3421}$-tableaux 
of type $e$ and shape $31$.
Four of these are
\begin{equation}\label{eq:rfrsstabs31}
\raisebox{-6mm}{\includegraphics[height=12mm]{tableaux/31_1432}}\ ,\quad
\raisebox{-6mm}{\includegraphics[height=12mm]{tableaux/31_2134}}\ ,\quad
\raisebox{-6mm}{\includegraphics[height=12mm]{tableaux/31_3214}}\ ,\quad
\raisebox{-6mm}{\includegraphics[height=12mm]{tableaux/31_4321}}\ ,
\end{equation}
where $\pi_i$ represents the unique path from source $i$ to sink $i$.
These tableaux $U$ of shape $31$
%and if we construct from each tableau $U$ 
%yield tableaux $U_2 \circ U_1$ of shape $4$,
yield tableaux $U_1 \circ U_2$ of shape $4$,
\begin{equation*}
%\label{eq:cyltabs4}
%\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_11433224}}\ ,\quad
%\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_22133441}}\ ,\quad
%\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_33122441}}\ ,\quad
%\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_44233112}}\ ,
\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_4321}}\ ,\quad
\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_1342}}\ ,\quad
\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_2143}}\ ,\quad
\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_3214}}\ ,
\end{equation*}
%which have $3$, $1$, $2$, and $4$ inversions, respectively.
which have $5$, $2$, $2$, and $3$ inversions, respectively.
Together, 
they
%the tableaux 
%contribute $q + q^2 + q^3 + q^4$ 
contribute $2q^2 + q^3 + q^5$ 
to $\psi_q^{31}(q_{e,3421} C'_{3421}(q)) = 1 + 3q + 5q^2 + 5q^3 + 3q^4 + q^5$.
%Similarly,
%The reader can verify that the same four tableaux of shape $31$
%yield tableaux $U_2 \circ U_1$ of shape $4$
%which have $3$, $1$, $2$, and $4$ inversions, respectively.
Now consider the sum in (\ref{eq:swqpsi2}).
It is easy to verify that there are six right-anchored 
row-semistrict $F_{3421}$-tableaux of type $e$ and shape $31$: 
the first and fourth tableaux
in (\ref{eq:rfrsstabs31}) and the four tableaux
%Four of these are
\begin{equation*}
%\label{eq:lacyltabs31}
\raisebox{-6mm}{\includegraphics[height=12mm]{tableaux/31_1342}}\ ,\quad
\raisebox{-6mm}{\includegraphics[height=12mm]{tableaux/31_2431}}\ ,\quad
\raisebox{-6mm}{\includegraphics[height=12mm]{tableaux/31_3421}}\ ,\quad
\raisebox{-6mm}{\includegraphics[height=12mm]{tableaux/31_4231}}\ .
\end{equation*}
These tableaux $U$ of shape $31$
%yield six tableaux $U_2 \circ U_1$ of shape $4$: 
yield six tableaux $U_1^R \circ U_2^R$ of shape $4$, 
%the 
%second and third tableaux in (\ref{eq:cyltabs4}) and the four 
%six tableaux
\begin{equation*}
%\label{eq:lacyltabs4}
\begin{gathered}
%\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_11233442}}\ ,\quad
%\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_11244332}}\ ,\quad
%\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_44122331}}\ ,\quad
%\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_44133221}}\ ,
\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_2341}}\ ,\quad
\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_1234}}\ ,\\
\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_2431}}\ ,\quad
\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_1342}}\ ,\quad
\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_1243}}\ ,\quad
\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_1324}}\ ,
\end{gathered}
\end{equation*}
%which have $1$, $2$, $0$, $1$, $2$, and $3$ inversions, respectively.
which have $2$, $0$, $3$, $2$, $1$, and $1$ inversions, respectively.
Together, 
%they
the six tableaux 
contribute $1 + 2q + 2q^2 + q^3$
%) = 1 + 3q + 5q^2 + 5q^3 + 3q^4 + q^5$.
to 
%$[3]_q [1]_q (1 + 2q + 2q^2 + q^3) = 1 + 3q + 5q^2 + 5q^3 + 3q^4 + q^5$.
%to 
%$\psi_q^{31}(q_{e,3421} C'_{3421}(q)) = 
$[3]_q [1]_q (1 + 2q + 2q^2 + q^3) 
%= 1 + 3q + 5q^2 + 5q^3 + 3q^4 + q^5
= \psi_q^{31}(q_{e,3421} C'_{3421}(q))$. 
%The reader can verify that the same six tableaux of shape $31$
%yield six different tableaux $U_1 \circ U_2$ of shape $4$
%which have $1$, $2$, $0$, $1$, $2$, and $3$ inversions, respectively.

We will state three more combinatorial formulas for 
$\psi_q^\lambda(\qew C'_w(q))$
in Theorems~\ref{t:qpsiO} and \ref{t:qpsi}.
To justify these,
%prove special cases of 
%the two formulas in Conjecture~\ref{conj:qpsi},
%the above conjectured 
%formulas for $\psi_q^\lambda(\qew C'_w(q))$,
%and to state a third formula,
% state and conjecture different
%more conjectured 
we 
%will 
associate a 
%particular 
polynomial 
$O(F) \in \mathbb N[q]$ 
to a descending star network $F$,
and a 
%particular 
path tableau $V(F,I)$ 
to the pair $(F,I)$, where $I$ is an ordered set partition 
of $[n]$ of type $\lambda$.

\begin{defn}\label{d:OFdef}
Let $F$ be a descending star network, and let
%ting
\begin{equation}\label{eq:concat2}
G_{[c_1,d_1]} \circ \cdots \circ G_{[c_t,d_t]}
\end{equation}
be the concatenation of star networks which corresponds to $F$
as in Section~\ref{s:dsn}.
Define the polynomial
$O(F) \in \mathbb N[q]$ by
\begin{equation}\label{eq:OFdef}
O(F) = \begin{cases}
\frac{\displaystyle{\prod_{i=1}^t}[d_i - c_i]_q!}
{\displaystyle
{
%\nTksp \nTksp \nTksp \nTksp \nTksp 
%\prodsb{[c_i,d_i] \prec \ntnsp \cdot [c_j,d_j]\\c_i < c_j}
%\nTksp \nTksp \nTksp \nTksp \nTksp \ntksp \ntnsp
%[d_j - c_i]_q!
\prodsb{[c_i,d_i] \prec \ntnsp \cdot [c_j,d_j]\\ c_i < c_j}
\nTksp \nTksp \ntksp \ntnsp
[d_i - c_j]_q!
\prodsb{[c_i,d_i] \prec \ntnsp \cdot [c_j,d_j]\\ c_j < c_i}
\nTksp \nTksp \ntksp \ntnsp
[d_j - c_i]_q!
%\prodsb{[c_i,d_i] \prec \ntnsp \cdot [c_j,d_j]\\ c_i < c_j < d_i < d_j}
%[d_i - c_j]_q!}
}}
&\text{if $F$ is connected},\\
0 &\text{if $F$ is disconnected}.
%&\text{if $F \neq F_e$ is connected},\\
%0 &\text{if $F \neq F_e$ is disconnected}.
%1 &\text{if $F = F_e$ is connected ($n = 1$)}.
\end{cases}
%O(F) = \begin{cases}
%\displaystyle{\frac{[d_1 - c_1]_q! \cdots [d_t - c_t]_q!}
%{[d_2 - c_1]_q! \cdots [d_t - c_{t-1}]_q!}} &\text{if $F$ is connected},\\
%0 &\text{otherwise}.
%\end{cases}
\end{equation}
%where $[a]_q! = [a]_q [a-1]_q \cdots [1]_q$ for $a \geq 1$ and $[0]_q! = 1$.
\end{defn}

For example, the connected descending star network $F_{3421}$ in (\ref{eq:F3421})
corresponds to the concatenation $G = G_{[2,4]} \circ G_{[1,3]}$
and two-element poset of intervals $[2,4] \precdot [1,3]$.  
Thus we have
\begin{equation*}
O(F_{3421}) 
= \frac{[4-2]_q! [3-1]_q!}{[3-2]_q!} 
= \frac{[2]_q! [2]_q!}{[1]_q!} 
= (1+q)^2.
\end{equation*}
Note that for the identity element $e \in \sn$ we have
$O(F_e) = 1$ if $n=1$ and $O(F_e) = 0$ otherwise.
%
Letting $\pi = (\pi_1, \dotsc, \pi_n)$ 
be the unique path family 
of type $e$ covering $F$, define
$V(F,I)$ to be the unique (row-semistrict) $\pi$-tableau of shape $\lambda$
for which $L(V(F,I))$ is a row-strict Young tableau 
containing indices $I_j$ in row $j$.
For $S$ a $k$-element subset of $[n]$, 
let $F|_S$ denote the zigzag network of order $k$ 
isomorphic to the subnetwork 
of $F$ 
covered by paths $\{\pi_i \,|\, i \in S \}$.



Also essential to our proofs of Theorems~\ref{t:qpsiO} and \ref{t:qpsi}
%($\star$ whatever)
%the equivalence of (\ref{eq:qpsi1}) and 
%the two equalities in (\ref{eq:qpsi2}) 
is a map 
%\begin{equation*}
\begin{multline*}
%\label{eq:iotadef1}
\iota: 
%\{ w \in \sn \,|\, w \text{ avoids $312$, $F_w$ connected } \}
\{ w \in \sn \,|\, w \text{ \avoidsp, $F_w$ connected}\,\} \\
\rightarrow
%%\{ w \in \mfs{n-1} \,|\, w \text{ avoids $312$, $F_w$ connected }\}.
\{ w \in \mfs{n-1} \,|\, w \text{ \avoidsp, $F_w$ connected}\,\}.
\end{multline*}
Let $F_w$ be a connected zig-zag network
%Assuming $F_w$ to be a connected descending star network 
of order $n \geq 2$
corresponding to the concatenation (\ref{eq:concat2}).
By Observation~\ref{o:firstorlast} we may assume that $t=1$,
or $d_1 = n$ and $[c_1, d_1] \precdot [c_2, d_2]$, 
or $d_t = n$ and $[c_{t-1}, d_{t-1}] \precdot [c_t, d_t]$.
%we necessarily have $d_1 = n$.
We declare
% avoid the pattern $312$
%($\star$ Introduce the map $\iota$.)
%Let $w \in \sn$ avoid the pattern $312$ and let the planar network
%$F_w$ of order $n$ correspond to the concatenation (\ref{eq:concat2})
%and the numbers $c_1, \dotsc, c_r, d_1, \dotsc, d_r$.
%Define a map $\iota: \sn \rightarrow \mfs{n-1}$ by declaring
$\iota(w)$ to be the permutation whose descending star network 
$F_{\iota(w)}$ of order $n-1$
is obtained from $F_w$
by deleting the path from source $n$ to sink $n$,
and, in the case that 
$d_1 = n$ and $d_2 = n-1$
($d_t = n$ and $d_{t-2} = n-1$),
by contracting one edge whose vertices correspond to the central vertices of 
$G_{[c_1,n]}$ and $G_{[c_2, n-1]}$
($G_{[c_{t-1},n-1]}$ and $G_{[c_t, n]}$).
Equivalently, $F_{\iota(w)}$ is the zig-zag network 
corresponding to the concatenation
\begin{equation}\label{eq:iotadef}
\begin{cases} 
G_{[c_1, n-1]} &\text{if $t = 1$},\\
G_{[c_2,d_2]} \circ \cdots \circ G_{[c_t,d_t]} 
&\text{if $d_1 = n$ and $d_2 = n-1$},\\
G_{[c_1,d_1]} \circ \cdots \circ G_{[c_{t-1},d_{t-1}]} 
&\text{if $d_t = n$ and $d_{t-1} = n-1$},\\
G_{[c_1,n-1]} \circ G_{[c_2,d_2]} \circ \cdots \circ G_{[c_t,d_t]} 
&\text{if $d_1 = n$ and $d_2 < n-1$},\\
G_{[c_1,d_1]} \circ \cdots \circ G_{[c_{t-1},d_{t-1}]} \circ G_{[c_t, n-1]} 
&\text{if $d_t = n$ and $d_{t-1} < n-1$}.
\end{cases}
\end{equation}

%(Since $F_w$ is connected, we necessarily have $d_1 = n$.)
%($\star$ Redo the above for general zig-zag networks?)

For example, let $n=6$. 
$F_{256431}$ corresponds to the concatenation
$G_{[3,6]} \circ G_{[2,5]} \circ G_{[1,2]}$, with $d_2 = 5 = n-1$.
Removing the path from source $6$ to sink $6$,
and contracting the edge whose endpoints correspond to the central vertices
of $G_{[3,6]}$ and $G_{[2,5]}$, 
we obtain $F_{\iota(256431)}$, which can be shown 
(as in the example preceding Theorem~\ref{t:lem53nndcb})
%as in the previous example
to be $F_{25431}$.  
\begin{equation}\label{eq:F256431}
F_{256431} = \raisebox{-10mm}{
%\includegraphics[height=12.25mm]{xstars3}}, \qquad
\includegraphics[height=22mm]{xfigures/iota1}}\ \mapsto
%\includegraphics[height=12.25mm]{xfigures/xstars3}}, \qquad
%w = \binom{1234}{3421}.
\raisebox{-10mm}{
\includegraphics[height=18mm]{xfigures/iota2}}\ \cong
\raisebox{-10mm}{
\includegraphics[height=18mm]{xfigures/iota3}}\
= F_{25431}.
%\end{center}
%}
%w = 3421.
%\thinspace \thinspace 
%\quad \qquad \qquad 
%\raisebox{-10mm}{$\square$}
%\end{gathered}
\end{equation}
Similarly,
$F_{246531}$ corresponds to the concatenation
$G_{[3,6]} \circ G_{[2,4]} \circ G_{[1,2]}$, with $d_2 = 4 < n-1$.
% = 5$.
Removing the path from source $6$ to sink $6$,
we obtain $F_{\iota(246531)}$, which can be shown 
to be $F_{24531}$.  
%($\star$ Give another description of $\iota(w)$ that doesn't involve planar networks?)
%($\star$ Give an example.)
\begin{equation}\label{eq:F246531}
%\begin{gathered}
%\label{eq:xfiguresextrax}
%\smash{
%\begin{center}
%\qquad \qquad \qquad
%F = \raisebox{-6mm}{
%\includegraphics[height=12mm]{xstars1}}, \qquad
%\includegraphics[height=12mm]{xfigures/xstars1}}, \qquad
F_{246531} = \raisebox{-10mm}{
\includegraphics[height=22mm]{xfigures/iota4}}\ \mapsto
%G = \raisebox{-6mm}{
%\includegraphics[height=12mm]{xstars2}}, \qquad
%\includegraphics[height=12mm]{xfigures/xstars2}}, \qquad
\raisebox{-10mm}{
\includegraphics[height=18mm]{xfigures/iota5}}\ 
= F_{24531}.
\end{equation}
%On the other hand, 

%($\star$ Is this even necessary? Probably delete this proposition
%and its proof.)
%While $\iota$ is not invertible, we do have the following.
%\begin{prop}\label{p:iotainverse}
%Let $F_u$ be a conneted zig-zag network of order $n-1 \geq 2$ corresponding
%to a concatenation (\ref{eq:concat2}) in which the unique interval
%containing $n-1$ is $[b,n-1]$ for some $b \leq n-2$.  
%If $[b, n-1]$ is minimal in the order $\preceq$, then 
%for each number $\ell \in [b,n-1]$ there is a unique connected zig-zag netowork
%$F_w$ of order $n$ such that $F_{\iota(w)} = F_v$ and in the concatenation
%(\ref{eq:concat2}) corresponding to $F_w$, the unique interval containing $n$
%is $[\ell,n]$.
%
%%For each pair $(F_v, \ell)$
%%where $F_v$ is a connected descending star network of order $n-1$ 
%%corresponding to a concatenation of the form
%%%(\ref{eq:concat2})
%%$G_{[k,n-1]} \circ \cdots$ for some $k \leq n-2$
%%% \circ G_{[c_t,d_t]}$
%%and $\ell$ satisfies $k \leq \ell \leq n-1$,
%%there is a unique connected descending star network $F_w$ of order $n$ 
%%such that $F_{\iota(w)} = F_v$
%%and $F_w$ corresponds to a concatenation of the form
%%$G_{[\ell,n]} \circ \cdots$.
%\end{prop}
%\begin{proof}
%Assume that $F_u$ is a connected zig-zag
%network corresponding
%to the concatenation (\ref{eq:concat2}), with 
%%$c_1 = k$, $d_1 = n-1$.
%$[k,n-1]$ the unique interval containing $n-1$.
%Now define $F_w$ to be the descending star network corresponding
%to the concatenation
%\begin{equation*}
%\begin{cases}
%G_{[k,n]} \circ G_{[c_2,d_2]} \circ \cdots \circ G_{[c_t,d_t]} &\text{if $\ell %= k$}\\
%G_{[\ell,n]} \circ G_{[c_1,d_1]} \circ \cdots \circ G_{[c_t,d_t]} &\text{if $\e%ll > k$}.
%\end{cases}
%\end{equation*}
%By the definition of $\iota$, we have in the first case 
%that $F_{\iota(w)}$ corresponds to the concatenation
%$G_{[k,n-1]} \circ G_{[c_2,d_2]} \circ \cdots \circ G_{[c_t,d_t]}$,
%and in the second case that $F_{\iota(w)}$ corresponds to the concatenation
%$G_{[c_1,d_1]} \circ \cdots \circ G_{[c_t,d_t]}$.
%\end{proof}

\ssec{Right-anchored, row-semistrict $F$-tableaux and $O(F)$}

%Let $w \in \sn$ \avoidp.  
Inversions in right-anchored, row-semistrict 
path tableaux are 
closely related to the polynomials
$\{ O(F) \,|\, F \text{ a zig-zag network of order } 1,\dotsc,n \}$ 
%$\{ O(F_u) \,|\, u \in \sm, m \leq n \}$. 
%$O(F)$ 
%defined 
in Definition~\ref{d:OFdef}.
%(\ref{eq:OFdef}).
In order to state this relationship precisely (Lemma~\ref{l:ranchO})
and state a third combinatorial formula for $\psi_q^\lambda(\qew C'_w(q))$
(Theorem~\ref{t:qpsiO}), 
we define a family of sets 
$\{Z(F) \,|\, F \text{ a zig-zag network} \}$ of tableaux
%$\{Z(F_u) \,|\, u \in \sm F \text{ a zig-zag network of order } 1,\dotsc,n \}$
and maps between these.
For $F$ a zig-zag network of order $m$, let $Z(F)$ be the set of
right-anchored, row-semistrict $F$-tableaux of type $e$ and shape $(m)$.
Note that if $F$ is not connected, then $Z(F) = \emptyset$, since a 
right-anchored $F$-tableau cannot be row-semistrict when $F$ is disconnected.
%Put Matt's material here.

Now let $F_w$ be a connected zig-zag network of order $m$ which corresponds
to the concatenation (\ref{eq:concat2}), and let $[b,m]$ be the unique interval
in (\ref{eq:concat2})
%this concatenation 
to contain $m$.
%in which the unique interval containing $m$ is $[b,m]$.
Define a map
\begin{equation*}
%\label{eq:gammadef}
\begin{aligned}
\gamma: Z(F_w) &\rightarrow Z(F_{\iota(w)}) \times \{ 0, 1, \dotsc, m-b-1 \}\\
U &\mapsto (U',k),
\end{aligned}
\end{equation*}
by declaring $U'$ to be the tableau obtained from $U$ by deleting $\pi_n$,
and by declaring $k$ to be the number of indices in the interval $[b,m-1]$
appearing to the left of $m$ in $L(U)$.

For example, consider the 
%zig-zag 
network $F_{256431}$ in (\ref{eq:F256431})
%$F_{43765821}$ 
%\begin{equation*}
%\text{figure},
%\end{equation*}
%which corresponds to the concatenation
%$G = G_{[6,8]} \circ G_{[3,7]} \circ G_{[1,4]}$,
and let $U$ be the tableau
\begin{equation*}
%\pi_7 \pi_5 \pi_3 \pi_2 \pi_8 \pi_6 \pi_4 \pi_1.
%\pi_4 \pi_5 \pi_6 \pi_3 \pi_2 \pi_1
\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/6_456321}}\ .
\end{equation*}
Then the unique interval in $G$ containing $m = 6$ is $[3,6]$, and there 
%element of this interval
are two indices in this interval 
appearing to the left of $6$ in $L(U)$.
%: $7$.
Thus $\gamma(U) = (U', 2)$ where $U'$ is the tableau
\begin{equation*}
%\pi_7 \pi_5 \pi_3 \pi_2 \pi_6 \pi_4 \pi_1.
\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/5_45321}}\ .
\end{equation*}

\begin{lem}\label{l:gamma}
For each connected zig-zag network $F_w$ of order $m$, the map $\gamma$
is a bijection.  Furthermore, if $\gamma(U) = (U', k)$ then 
$\inv(U^R) = \inv(U'^R) + k$. 
\end{lem}
\begin{proof}
To see that $\gamma$ is well-defined, fix $U \in Z(F_w)$ and let
$L(U) = (i_1, \dotsc, i_m = 1)$, where $i_j = m$ and $j < m$.
Clearly $U'$ is right-anchored. If $j = 1$ then 
$U' = (\pi_{i_2}, \dotsc, \pi_{i_m})$
is row-semistrict.  If $j \geq 1$, then 
$U' = (\pi_{i_1}, \dotsc, \pi_{i_{j-1}}, \pi_{i_{j+1}}, \dotsc, \pi_{i_m})$ 
is also row-semistrict,
since $\pi_m \not >_{P(F_w)} \pi_{i_{j+1}}$ implies that 
$\pi_{i_{j-1}} \not >_{P(F_{\iota(w)})} \pi_{i_{j+1}}$.

To invert $\gamma$, find an entry $i_d$ of $L(U')$ which belongs to
$[b,m-1]$ and has exactly $k$ 
indices from the interval $[b,m-1]$ to its left.
(This is possible, since $0 \leq k \leq m - b - 1$.)
Now create a new $F_w$-tableau by inserting $\pi_m$
%the letter $m$ 
into $U'$
%$L(U')$ 
immediately before $\pi_{i_d}$.
This map is well-defined, because $\pi_{i_d}$ intersects $\pi_m$.
It is clear that the map inverts $\gamma$.  
%The inverse map 
% and define $U$ to be the unique
%right-anchored, row-semistrict $F_w$-tableau satisfying $L(U) = T$.
%in $P(F_w)$ than the path
Since $U$ has type $e$, it is clear that the number of inversions in $U^R$
involving $\pi_m$ is equal to the number of indices in the interval $[b, m-1]$
appearing to the left of $m$ in $L(U)$.  
It follows that $\inv(U^R) = \inv(U'^R) + k$.
\end{proof}


By the above lemma, we can interpret $O(F)$ as a generating function
for inversions in tableaux belonging to $Z(F)$.
\begin{lem}\label{l:ranchO}
For each $w \in \mfs m$ \avoidingp, we have
\begin{equation}\label{eq:ranchO}
\sum_{U \in Z(F_w)} q^{\inv(U^R)} = O(F_w).
\end{equation}
\end{lem}
\begin{proof}
When $m = 1$, both sides of (\ref{eq:ranchO}) are $1$.
Now assume that (\ref{eq:ranchO}) holds for all zig-zag networks
corresponding to \pavoiding permutations in $\mfs 1, \dotsc, \mfs{m-1}$,
and consider $w \in \mfs m$ \avoidingp.  Let $F_w$ correspond to the 
concatenation (\ref{eq:concat2}).
%$G_{[c_1,d_1]} \circ \cdots \circ G_{[c_t,d_t]}$.
If $F_w$ is disconnected, then $Z(F_w) = \emptyset$ and both sides of 
(\ref{eq:ranchO}) are $0$.
If $F_w$ is connected, let $[b,m]$ be the unique interval in (\ref{eq:concat2})
%the concatenation
to contain $m$.  
By induction and Lemma~\ref{l:gamma} we have
\begin{equation}\label{eq:Oinduction}
\sum_{U \in Z(F_w)} q^{\inv(U^R)} = 
\sum_{k=0}^{m-b-1} q^k \sum_{U' \in Z(F_{\iota(w)})} q^{\inv(U'^R)}
= [m-b]_q O(F_{\iota(w)}).
\end{equation}
By Observation~\ref{o:firstorlast} we may assume that we have 
$t=1$, or
$[b,m] = [c_1,d_1] \precdot [c_2,d_2]$, 
%$[c_1,d_1] \precdot [c_2,d_2]$ 
or $[c_{t-1},d_{t-1}] \precdot [c_t,d_t] = [b,m]$.
In the first case, the expression (\ref{eq:Oinduction}) is
%$[m-b]_q[m-b-1]_q! = 
$[m-b]_q!$.
In the second case,
%Now 
by Definition~\ref{d:OFdef}
%(\ref{eq:OFdef}) 
and (\ref{eq:iotadef}), it is
%this is equal to
%($\star$ Check for too many or too few intervals included in
%denominator products.)
% we have
\begin{equation}\label{eq:OandO}
%[n-b]_q O(F_{\iota(w)}) = 
\begin{cases}
[m-c_1]_q 
\frac{
{\displaystyle[m-1-c_1]_q! [d_2-c_2]_q! \cdots [d_t-c_t]_q!}}
{\displaystyle
{\prodsb{[c_i,d_i] \prec \ntnsp \cdot [c_j,d_j]\\ c_i < c_j}
\nTksp \nTksp \ntksp \ntnsp
[d_i - c_j]_q!
\prodsb{[c_i,d_i] \prec \ntnsp \cdot [c_j,d_j]\\ c_j < c_i}
\nTksp \nTksp \ntksp \ntnsp
[d_j - c_i]_q!}}
&\text{if $d_1 = m$, $d_2 < m-1$},\\
\displaystyle{
\frac{[m-c_1]_q!}{[d_2 - c_1]_q!} 
}
\frac{
[d_2-c_2]_q! \cdots [d_t-c_t]_q!}
{\displaystyle{
\prodsb{[c_i,d_i] \prec \ntnsp \cdot [c_j,d_j]\\ c_i < c_j}
\nTksp \nTksp \ntksp \ntnsp
[d_i - c_j]_q!
\prodsb{[c_i,d_i] \prec \ntnsp \cdot [c_j,d_j]\\ c_j < c_i \\ i \geq 2}
\nTksp \nTksp \ntksp \ntnsp
[d_j - c_i]_q!}}
&\text{if $d_1 = m$, $d_2 = m-1$}.
%,\\
%[m-c_t]_q 
%\frac{
%{\displaystyle[d_2-c_2]_q! \cdots [d_{t-1}-c_{t-1}]_q![m-1-c_t]_q! }}
%{\displaystyle
%{\prodsb{[c_i,d_i] \prec \ntnsp \cdot [c_j,d_j]\\ c_i < c_j}
%\nTksp \nTksp \ntksp \ntnsp
%[d_i - c_j]_q!
%\prodsb{[c_i,d_i] \prec \ntnsp \cdot [c_j,d_j]\\ c_j < c_i}
%\nTksp \nTksp \ntksp \ntnsp
%[d_j - c_i]_q!}}
%&\text{if $d_t = m$, $d_{t-1} < m-1$},\\
%\displaystyle{
%\frac{[m-c_t]_q!}{[d_{t-1} - c_t]_q!} 
%}
%\frac{
%[d_2-c_2]_q! \cdots [d_t-c_t]_q!}
%{\displaystyle{
%\prodsb{[c_i,d_i] \prec \ntnsp \cdot [c_j,d_j]\\ c_i < c_j}
%\nTksp \nTksp \ntksp \ntnsp
%[d_i - c_j]_q!
%\prodsb{[c_i,d_i] \prec \ntnsp \cdot [c_j,d_j]\\ c_j < c_i}
%\nTksp \nTksp \ntksp \ntnsp
%[d_j - c_i]_q!}}
%&\text{if $d_t = m$, $d_{t-1} = m-1$}.
\end{cases}
\end{equation}
In the third case, we obtain an expression similar to (\ref{eq:OandO}).
In all cases, the expression 
%this 
is equal to $O(F_w)$.
\end{proof}

Now we can state the precise relationship between the polynomials
$O(F)$ and inversions in right-anchored, row-semistrict path tableaux.

\begin{thm}\label{t:qpsiO}
Let $w \in \sn$ \avoidp.
For $\lambda = (\lambda_1, \dotsc, \lambda_r) \vdash n$, we have
\begin{equation}\label{eq:qpsi3}
\psi_q^\lambda(\qew C'_w(q)) = 
[\lambda_1]_q \cdots [\lambda_r]_q
\sumsb{I \vdash [n]\\ \type(I) = \lambda}
q^{\inv(V(F_w,I)_1 \circ \cdots \circ V(F_w,I)_r)} O(F_w|_{I_1}) \cdots O(F_w|_{I_r}).
%&= [\lambda_1]_q \cdots [\lambda_r]_q
%\sumsb{I \vdash [n]\\ \type(I) = \lambda}
%q^{\inv(V(F_w,I)_r \circ \cdots \circ V(F_w,I)_1)} O(F_w|_{I_1}) \cdots O(F_w|_%{I_r}).
%\end{aligned}
\end{equation}
\end{thm}
\begin{proof}
By Theorem~\ref{t:swqpsi}, $\psi_q^\lambda(\qew C'_w(q))$
is equal to 
%the sum on 
the right-hand side of (\ref{eq:swqpsi2}).
Grouping terms in the sum and using (\ref{eq:invdecomp}),
we may rewrite this expression as
%the sum there as
%right-hand side of (\ref{eq:swqpsi1}) as
\begin{equation}\label{eq:qpsi4}
[\lambda_1]_q \cdots [\lambda_r]_q
\nTksp \ntksp \sumsb{I \vdash [n]\\ \type(I) = \lambda} \nTksp
\sum_U q^{\inv(U_1^R \circ \cdots \circ U_r^R)} = 
[\lambda_1]_q \cdots [\lambda_r]_q
\nTksp \ntksp \sumsb{I \vdash [n]\\ \type(I) = \lambda} \nTksp
\sum_U q^{\inv(U_1^R) + \cdots + \inv(U_r^R) + \inv(U^\tr)},
\end{equation}
where 
%for each ordered set partition $I = (I_1, \dotsc, I_r)$,
$U$ now varies over the subset of right-anchored, row-semistrict
tableaux of type $e$ and shape $\lambda$ satisfying
$U_j = I_j$ for 
%$j = 1, \dotsc, r$.
each component of the appropriate ordered set partition
$I = (I_1,\dotsc,I_r)$.  
For fixed $I$, this inner sum can be rewritten as
\begin{equation*}
%\label{eq:qpsi5}
\sum_{W^{(1)}}q^{\inv((W^{(1)})^R)} \cdots \sum_{W^{(r)}}q^{\inv((W^{(r)})^R)}
\sum_U q^{\inv(U^\tr)},
\end{equation*}
where $W^{(j)}$ varies over right-anchored, row-semistrict $F_w|_{I_j}$-tableaux
of shape $\lambda_j$ and type $e \in \mfs{\lambda_j}$,
and $U$ again varies as in (\ref{eq:qpsi4}).
%over right-anchored, row-semistrict tableaux of
%shape $\lambda$ and type $e \in \mfs m$.  
By Lemma~\ref{l:ranchO}, the first $r$ sums are equal to
$O(F_w|_{I_1}), \dotsc, O(F_w|_{I_r})$, and it
is easy to see that for any tableau $U$ in the last sum, we have
$\inv(U^\tr) = \inv(V(F_w,I)_1 \circ \cdots \circ V(F_w,I)_r)$.
Thus we obtain
%by Lemma~\ref{l:ranchO}, 
%(\ref{eq:qpsi5}) is equal to
the right-hand side of (\ref{eq:qpsi3}).
\end{proof}

%In Theorem~\ref{t:qpsiO} we will prove that
%for $\lambda = (\lambda_1, \dotsc, \lambda_r) \vdash n$
%and $w \in \sn$ \avoidingp, we have
%\begin{equation}\label{eq:qpsi3}
%\psi_q^\lambda(\qew C'_w(q)) 
%= [\lambda_1]_q \cdots [\lambda_r]_q
%\sumsb{I \vdash [n]\\ \type(I) = \lambda}
%q^{\inv(V(F_w,I)_1 \circ \cdots \circ V(F_w,I)_r)} 
%O(F_w|_{I_1}) \cdots O(F_w|_{I_r}).
%\end{equation}

For example, consider again the descending star network 
$F_{3421}$ in (\ref{eq:F3421}) and the expression in (\ref{eq:qpsi3}).
%Now consider the first 
%consider the sum in (\ref{eq:qpsi3}).
The ordered set partitions of $[4]$
of type $31$ are $123|4$, $124|3$, $134|2$, and $234|1$.
Corresponding to the set partitions $I$ are the tableaux $V(F_{3421},I)$
of shape 31
\begin{equation*}\label{eq:lacyltabs31}
\raisebox{-6mm}{\includegraphics[height=12mm]{tableaux/31_4123}}\ ,\quad
\raisebox{-6mm}{\includegraphics[height=12mm]{tableaux/31_3124}}\ ,\quad
\raisebox{-6mm}{\includegraphics[height=12mm]{tableaux/31_2134}}\ ,\quad
\raisebox{-6mm}{\includegraphics[height=12mm]{tableaux/31_1234}}\ ,
\end{equation*}
respectively,
where $(\pi_1, \pi_2, \pi_3, \pi_4)$ is the unique path family of type $e$
covering $F_{3421}$.  
%These in turn yield tableaux $V(F_{3421},I)_2 \circ V(F_{3421},I)_1$ of shape 4
These in turn yield tableaux $V(F_{3421},I)_1 \circ V(F_{3421},I)_2$ of shape 4
\begin{equation*}
%\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_4123}}\ ,\quad
%\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_3124}}\ ,\quad
%\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_2134}}\ ,\quad
%\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_1234}}\ ,
\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_1234}}\ ,\quad
\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_1243}}\ ,\quad
\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_1342}}\ ,\quad
\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_2341}}\ ,
%q^{\inv(\pi_4\pi_1\pi_2\pi_3)} = q^2,\quad
%q^{\inv(\pi_3\pi_1\pi_2\pi_4)} = q^2,\quad
%q^{\inv(\pi_2\pi_1\pi_3\pi_4)} = q,\quad
%q^{\inv(\pi_1\pi_2\pi_3\pi_4)} = 1.
\end{equation*}
%having $2$, $2$, $1$, and $0$ inversions, respectively.
having $0$, $1$, $2$, and $2$ inversions, respectively.
The subnetworks 
$F_{3421}|_{123}$,
$F_{3421}|_{124}$,
$F_{3421}|_{134}$, 
$F_{3421}|_{234}$ and polynomials
$O(F_{3421}|_{123})$,
$O(F_{3421}|_{124})$,
$O(F_{3421}|_{134})$, 
$O(F_{3421}|_{234})$ 
are
\begin{equation*}
\begin{alignedat}{4}
\raisebox{-8mm}{\includegraphics[height=16mm]{xfigures/xstars1nwo4}}\ 
\ntksp \ntnsp &\cong \ntksp
\raisebox{-6mm}{\includegraphics[height=12mm]{xfigures/xstars1nwo4s}},&\qquad
\raisebox{-8mm}{\includegraphics[height=16mm]{xfigures/xstars1nwo3}}\ 
\ntksp \ntnsp \cong \ntksp
\raisebox{-6mm}{\includegraphics[height=12mm]{xfigures/xstars1nwo3s}}&,&\qquad
\raisebox{-8mm}{\includegraphics[height=16mm]{xfigures/xstars1nwo2}}\ 
\ntksp \ntnsp \cong \ntksp
\raisebox{-6mm}{\includegraphics[height=12mm]{xfigures/xstars1nwo3s}}&,&\qquad
\raisebox{-8mm}{\includegraphics[height=16mm]{xfigures/xstars1nwo1}}\ 
\ntksp \ntnsp &\cong \ntksp
\raisebox{-6mm}{\includegraphics[height=12mm]{xfigures/xstars1nwo4s}},\\
%\end{equation*}
[3-1]_q! &= 1 + q, &
\frac{[3-2]_q! [2-1]_q!}{[2-2]_q!} = 1&, &
\frac{[3-2]_q! [2-1]_q!}{[2-2]_q!} = 1&, &
[3-1]_q! &= 1 + q.
%and satisfy
%\begin{equation*}
%\begin{gathered}
%O(F_{3421}|_{123}) = 
%O(F_{3421}|_{234}) = [3-1]_q! = 1 + q, \\
%O(F_{3421}|_{124}) = 
%O(F_{3421}|_{134}) = \frac{[3-2]_q! [2-1]_q!}{[2-2]_q!} = 1.
%\end{gathered}
\end{alignedat}
\end{equation*}
On the other hand, each subnetwork $F_{3421}|_{i}$ is simply a path and satisfies
$O(F_{3421}|_{i}) = 1$.  
Thus the four set partitions contribute
$q^0(1+q)(1)$, $q^1(1)(1)$,  $q^2(1)(1)$, $q^2(1+q)(1)$, or a total of
%$1(1+q)(1)$, $q(1)(1)$,  $q^2(1)(1)$, $q^2(1+q)(1)$, or a total of
$1 + 2q + 2q^2 + q^3$ to
$[3]_q[1]_q ( 1 + 2q + 2q^2 + q^3 ) = \psi_q^{31}(q_{e,3421} C'_{3421}(q))$.
%The reader can verify that the same four tableaux $V(F_{3421},I)$
%of shape $31$
%yield four different tableaux $V(F_{3421},I)_2 \circ V(F_{3421},I)_1$ of shape $4$
%which have $2$, $2$, $1$, and $0$ inversions, respectively.





%while each subnetwork $F_{3421}|_{i}$ 
%satisfies
%$O(F_{3421}|_{i}) = 1$.
%The subnetworks 
%$F_{3421}|_{123}$,
%$F_{3421}|_{124}$,
%$F_{3421}|_{134}$, and
%$F_{3421}|_{234}$ are
%\begin{equation*}
%\raisebox{-8mm}{\includegraphics[height=16mm]{xfigures/xstars1nwo4}}\ \cong
%\raisebox{-6mm}{\includegraphics[height=12mm]{xfigures/xstars1nwo4s}}\ ,\quad
%\raisebox{-8mm}{\includegraphics[height=16mm]{xfigures/xstars1nwo3}}\ \cong
%\raisebox{-6mm}{\includegraphics[height=12mm]{xfigures/xstars1nwo3s}}\ ,\quad
%\raisebox{-8mm}{\includegraphics[height=16mm]{xfigures/xstars1nwo2}}\ \cong
%\raisebox{-6mm}{\includegraphics[height=12mm]{xfigures/xstars1nwo3s}}\ ,\quad
%\raisebox{-8mm}{\includegraphics[height=16mm]{xfigures/xstars1nwo1}}\ \cong
%\raisebox{-6mm}{\includegraphics[height=12mm]{xfigures/xstars1nwo4s}}\ ,
%\end{equation*}
%and satisfy
%\begin{equation*}
%\begin{gathered}
%O(F_{3421}|_{123}) = 
%O(F_{3421}|_{234}) = [3-1]_q! = 1 + q, \\
%O(F_{3421}|_{124}) = 
%O(F_{3421}|_{134}) = \frac{[3-2]_q! [2-1]_q!}{[2-2]_q!} = 1.
%\end{gathered}
%\end{equation*}
%On the other hand, each subnetwork $F_{3421}|_{i}$ is simply a path and satisfies
%$O(F_{3421}|_{i}) = 1$.  
%Thus the four set partitions contribute
%$q^2(1+q)(1)$, $q^2(1)(1)$,  $q(1)(1)$, $1(1+q)(1)$, or a total of
%$1 + 2q + 2q^2 + q^3$ to
%$[3]_q[1]_q ( 1 + 2q + 2q^2 + q^3 ) = \psi_q^{31}(q_{e,3421} C'_{3421}(q))$.

%The similarity of Equations 
%(\ref{eq:qpsi1}) and (\ref{eq:qpsi2}) 
%to 
%(\ref{eq:swqpsi1}) and (\ref{eq:swqpsi2}) 
%is somewhat striking.  Indeed,
%we conjecture that they are all equivalent.
%\begin{conj}
%Conjectures \ref{conj:swqpsi} and \ref{conj:qpsi} are equivalent.
%\end{conj}

The special case $\lambda = (n)$ of Theorem~\ref{t:qpsiO} confirms a 
conjecture of Haiman~\cite[Conj.\,4.1]{HaimanHecke} concerning 
evaluations of the form $\phi_q^{(n)}(\qew C'_w(q)) = \psi_q^{(n)}(\qew C'_w(q))$.
\begin{prop}
Let $w = w_1 \cdots w_n \in \sn$ avoid the pattern $312$ 
and define the sequence $(f(1), \dotsc, f(n))$ 
by $f(j) = \max \{ w_1, \dotsc, w_j \}$.
Then we have
\begin{equation}\label{eq:qpsiH}
\psi_q^{(n)}(\qew C'_w(q)) = 
[n]_q [f(1) - 1]_q [f(2) - 2]_q \cdots [f(n-1) - (n-1)]_q.
\end{equation}
\end{prop}
\begin{proof}
Setting $\lambda = (n)$ in Theorem~\ref{t:qpsiO}, we have
\begin{equation}\label{eq:qpsiOn}
\psi_q^{(n)}(\qew C'_w(q)) = [n]_q q^0 O(F_w).
\end{equation}
If $F_w$ is not connected, then both sides of (\ref{eq:qpsiOn}) are $0$,
and for some index $k$ the prefix $w_1 \cdots w_k$ of $w$ belongs to $\mfs{k}$.
Thus $f(k) = k$ and the right-hand side of (\ref{eq:qpsiH}) is $0$ as well.

Assume therefore that $F_w$ is connected. 
Since $w$ avoids the pattern $312$, $F_w$ is a descending star network
and the intervals in the corresponding concatenation (\ref{eq:concat2})
%satisfy 
form the chain
%\begin{equation*}
%form a chain 
$[c_1, d_1] \precdot \cdots \precdot [c_t, d_t]$
with $d_1 = n$, $c_t = 1$.
%\end{equation*}  
Thus the formula (\ref{eq:OFdef}) for $O(F_w)$ becomes
%\begin{equation}\label{eq:DSNOFw}
\begin{multline*}\label{eq:DSNOFw}
[d_1-c_1]_q!
\frac{[d_2-c_2]_q!}{[d_2-c_1]_q!} \cdots \frac{[d_t-c_t]_q!}{[d_t-c_{t-1}]_q!}
= 
%\\
%\begin{gathered}
%{}
\Big([d_1-c_1]_q[d_1-(c_1+1)]_q \cdots [d_1-(n-1)]_q \Big) \cdot \\
%\\
\Big([d_2-c_2]_q[d_2-(c_2+1)]_q \cdots [d_2-(c_1-1)]_q \Big) 
%\cdot \\
%\vdots \\
\cdots
\Big([d_t-1]_q[d_t-2]_q \cdots [d_t-(c_{t-1}-1)]_q \Big).
%\end{gathered}
%\end{equation}
\end{multline*}
Defining $g(j) = \min \{ i \,|\, j \in [c_i,d_i] \}$ for $j = 1,\dotsc,n$,
we may now rewrite (\ref{eq:qpsiOn}) as
\begin{equation}\label{eq:almostHaiman}
\psi_q^{(n)}(\qew C'_w(q)) = [n]_q 
[d_{g(1)} - 1]_q [d_{g(2)} - 2]_q \cdots [d_{g(n-1)} - (n-1)]_q.
\end{equation}

Finally we claim that $d_{g(j)} = f(j)$ for $j = 1,\dotsc,n-1$.
We have $f(j) \leq d_{g(j)}$ 
%for each $j$ 
because there are no paths
in $F_w$ from source $j$ to sinks $d_{g(j)}+1, \dotsc, n$
and therefore by Observation~\ref{o:sourcetosink}, 
no paths from sources $1, \dotsc, j-1$ to these sinks either.
Similarly, we have $f(j) \geq d_{g(j)}$ because $j$ belongs to the interval
$[c_{g(j)}, d_{g(j)}]$ and $w_{c_{g(j)}} = d_{g(j)}$.
\end{proof}
It is straightforward to show that the right-hand side of 
(\ref{eq:almostHaiman}) coincides with the expression in 
\cite[Thm.\,7.1]{SWachsChromQ} for $(-1)^{n-1}n = (-1)^{n-\ell(n)} z_{(n)}$ 
times the coefficient of $p_n$ in $X_{P,q}$.

\ssec{Cylindrical $F$-tableaux and $O(F)$}

In Theorem~\ref{t:qpsi},
%\ref{t:qpsi12equiv},
we will prove an analog of Theorem~\ref{t:swqpsi}
in which sums are taken over (left-anchored) cylindrical $F$-tableaux.
To do so, we partition the set of cylindrical $F$-tableaux into
equivalence classes as follows.
Fix a permutation $w \in \sn$ \avoidingp, 
an integer partition $\lambda \vdash n$, 
and an ordered set partition
$I = (I_1, \dotsc, I_r)$ of $[n]$ of type $\lambda$.
Let $\mathcal C(I,F_w)$ be the set of cylindrical
$F_w$-tableaux $U$ such that for $j = 1,\dotsc,r$,
the set of entries of $L(U_j)$ is equal to $I_j$.
Let $\mathcal C_L(I,F_w)$ be the subset of these tableaux which
are left-anchored.
Now the cylindrical analogs of the sums in (\ref{eq:swqpsi1}) 
and (\ref{eq:swqpsi2}) are
%We now verify the truth of two components of Conjecture~\ref{conj:qpsi}:
%the second equality in (\ref{eq:qpsi2}), and the equivalence of
%(\ref{eq:qpsi1}) and (\ref{eq:qpsi2}). 
%% which we already know to be true are
%First, for each zig-zag network $F_w$, let us 
%partition the set of cylindrical $F_w$-tableaux of shape $\lambda$
%into 
%rewrite the sums over tableaux $U$ 
%%on the right-hand sides of
%%Equations 
%in (\ref{eq:qpsi1}), (\ref{eq:qpsi2}) 
%by collecting terms
%in which the sequence of sets of elements of the one-rowed tableaux 
%$L(U_1), \dotsc, L(U_r)$
%form a fixed ordered set partition $I = (I_1,\dotsc, I_r)$ of $[n]$.
%Given such an ordered set partition $I$ of shape $\lambda$ and a
%\pavoiding permutation $w$, 
%%$X = (X_1,\dotsc, X_r)$ of $[n]$.
%%\begin{prop}
%%Let $w \in \sn$ \avoidp.  
%%For $\lambda \vdash n$, the sum on the right-hand side of 
%%Equation (\ref{eq:qpsi2}) is equal to
%% we have
%Thus 
%we have that the sums
%$\sum_U q^{\inv(U_r \circ \cdots \circ U_1)}$
%in (\ref{eq:qpsi1}) and (\ref{eq:qpsi2})
%over cylindrical $F_w$-tableaux 
%and left-anchored cylindrical $F_w$-tableaux, respectively, 
%are equal to
\begin{equation}\label{eq:sumxu}
\begin{gathered}
\sum_U q^{\inv(U_1 \circ \cdots \circ U_r)} = 
\sumsb{I \vdash [n]\\ \type(I) = \lambda} \sum_{U \in \mathcal C(I,F_w)}
 q^{\inv(U_1 \circ \cdots \circ U_r)},\\
%\qquad
%\sum_U q^{\inv(U_r \circ \cdots \circ U_1)} =
\sum_U q^{\inv(U_1 \circ \cdots \circ U_r)} = 
\sumsb{I \vdash [n]\\ \type(I) = \lambda} \sum_{U \in \mathcal C_L(I,F_w)}
 q^{\inv(U_1 \circ \cdots \circ U_r)},
\end{gathered}
\end{equation}
where the left-hand sums are over cylindrical $F_w$-tableaux of shape $\lambda$
and left-anchored cylindrical $F_w$-tableaux of shape $\lambda$,
respectively.
%where the first sum is over
%ordered set partitions $I$ of type $\lambda$.
%and 
%(left-anchored) 
%cylindrical $F_w$-tableaux $U$ such that for $j = 1,\dotsc,r$,
%the set of entries of $L(U_j)$ is equal to $I_j$.
%\end{prop}
In both cases,
%the anchored and unanchored cases, 
it is easy to show that the 
inner right-hand sum factors as in Theorem~\ref{t:qpsiO}.
%a power of $q$ 
%times a product of generating functions of the form $O(F)$.
%%corresponding to the $r$ blocks of $I$.
To state this factorization explicitly, 
we relate $\inv(U_1 \circ \cdots \circ U_r)$ to intervals
in the concatenation (\ref{eq:concat2}).
\begin{lem}\label{l:indexpairs}
Let $w \in \sn$ \avoidp, and let $[c_1,d_1], \dotsc, [c_t,d_t]$ be
the intervals appearing in the concatenation (\ref{eq:concat2}) 
%be the concatenation
of star networks that corresponds to $F_w$.  Let $U$ be a cylindrical 
$F_w$-tableau having $r$ rows, and fix indices $p_1 < p_2$ in $[r]$.
Then we have
%\begin{equation*}
\begin{multline}\label{eq:indexpairs}
%\#\{ (\pi_a,\pi_b) \in U_{p_1} \times U_{p_2} \,|\, 
%(\pi_b, \pi_a) \text{ an inversion in } U_r \circ \cdots \circ U_1 \}\\
\#\{ (\pi_a,\pi_b) \in U_{p_2} \times U_{p_1} \,|\, 
(\pi_b, \pi_a) \text{ an inversion in } U_1 \circ \cdots \circ U_r \}\\
%\#\{ (\pi_a,\pi_b) \in U_{p_2} \times U_{p_1} \,|\, 
%\#\{ (\pi_a,\pi_b) \,|\, 
%(\pi_b, \pi_a) \text{ an inversion in } U_{p_1} \circ U_{p_2} \}
%(\pi_a, \pi_b) \text{ intersect }\}\\
=
\#\{ (a,b) \in L(U)_{p_2} \times L(U)_{p_1} \,|\, 
c_j \leq a < b \leq d_j \text{ for some } j\}.
\end{multline}
%\end{equation*}
\end{lem}
\begin{proof}
Let $A$ and $B$ denote the sets on the left- and right-hand sides of
(\ref{eq:indexpairs}), respectively.  Define a map $\varphi: A \rightarrow B$
as follows, assuming $(\pi_a, \pi_b) \in A$.
If $a$ and $b$ belong to a common interval $[c_i,d_i]$, then set
$\varphi((\pi_a, \pi_b)) = (a,b)$.  Otherwise, read $U_{p_1}$ cyclically from
left to right starting at $\pi_b$, and let $\pi_c$ be the first path which
lies entirely above $\pi_a$.  Then set $\varphi((\pi_a,\pi_b)) = (a,c)$.
We claim that $\varphi$ is a bijection.

To see that $\varphi$ is well defined, suppose that $a$ and $b$ belong to
no common interval $[c_i, d_i]$,
% in (\ref{eq:concat2}), 
and let $\pi_f$ be the path in 
$U_{p_1}$ terminating at sink $b$.  If $\pi_f$ intersects $\pi_a$, then
there exists a path in $F_w$ from source $a$ to sink $b$.
Since $a < b$, Observation~\ref{o:sourcetosink} and the comment following it
imply that $a$ and $b$ belong to a common interval $[c_i, d_i]$,
%in (\ref{eq:concat2}),
a contradiction.  Thus the set of paths lying strictly above $\pi_a$ in $U_{p_1}$
is nonempty, and the path $\pi_c$ is well defined.
% above satisfies $c > a$.
Suppose $a$ and $c$ belong to no common interval $[c_i,d_i]$,
% in (\ref{eq:conca2t}), 
and let $\pi_g$ be the path in $U_{p_1}$ cyclically preceding $\pi_c$.
% terminating at sink $c > a$.
Then $\pi_g$ terminates at sink $c > a$.
%, up to cyclic rotation, $U_{p_2}$ has the form
%\[\ytableausetup{mathmode}
%\begin{ytableau}
%\cdots & \pi_b & \cdots & \pi_g & \pi_c & \cdots
%\end{ytableau}\,
%\raisebox{1.5mm}{.}\]
By 
%by 
our choice of $\pi_c$, the path $\pi_g$ must intersect $\pi_a$.
%By Lemma~\ref{l:dsnintersect}, 
Thus there is a path in $F_w$ from source $a$ to sink $c$.  
But this contradicts Observation~\ref{o:sourcetosink}.

The inverse $\xi$ of $\varphi$ may be described as follows, 
assuming $(a,b) \in B$.
If $\pi_a$ intersects $\pi_b$,
% \neq \emptyset$, 
then set 
$\xi((a,b)) = (\pi_a, \pi_b)$.  Otherwise, read $U_{p_1}$ cyclically
from right to left starting at $\pi_b$, and let $\pi_c$ be the first path
such that $a$ and $c$ belong to no common interval $[c_i, d_i]$, and $c > a$.
Then set $\xi((a,b)) = (\pi_a, \pi_c)$.

To see that $\xi$ is well defined, suppose that $\pi_a$ does not intersect
$\pi_b$ and that $\xi((a,b)) = (\pi_a, \pi_c)$.  
Let $d$ be the index of the sink of $\pi_c$, so that $\pi_d$ 
% terminate at sink $d$.
%then $\pi_d$ 
immediately follows $\pi_c$.  By our choice of $\pi_c$, 
we have that $a$ and $d$ belong to a common interval $[c_i, d_i]$ 
or that $d < a$.  Thus by Observation~\ref{o:sourcetosink},
there is a path in $F_w$ from source $a$ to sink $d$,
and by Lemma~\ref{l:dsnintersect} paths $\pi_a$ and $\pi_c$ intersect.

Now we claim that $\varphi$ and $\xi$ are in fact inverse to one another.
This is clear when we restrict to pairs $(a,b)$ 
belonging to a common interval in (\ref{eq:concat2})
such that the paths $\pi_a$, $\pi_b$ intersect.
%and to pairs $(\pi_a, \pi_b)$ of intersecting paths.
Suppose therefore that $\pi_a$ intersects $\pi_b$, 
and that $a$, $b$ belong to no common interval in (\ref{eq:concat2}). 
Let $\varphi((\pi_a, \pi_b)) = (a,c)$
and let $\xi((a,c)) = (\pi_a, \pi_{b'})$.
Since $\pi_c$ lies entirely above $\pi_a$, we have $b' \neq c$.
%($\star$ or what is the reason for mentioning that $\pi_c$ lies entirely
%above $\pi_a$?)
Suppose $b' \neq b$ and let $\pi_f$ be the path in $U_{p_1}$ terminating at
sink $b'$.  
By the definitions of $\varphi$ and $\xi$, and since $a$, $b$ belong to
no common interval in (\ref{eq:concat2}), the row
$U_{p_1}$ (up to cyclic rotation) has the form
\begin{equation*}
\cdots\
\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/1_b}}\ 
\cdots\
\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/2_fbprime}}\ 
\cdots\
\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/1_c}}\ . 
\end{equation*}
%\[\ytableausetup{mathmode}
%\begin{ytableau}
%\cdots & \pi_b & \cdots & \pi_f & \pi_{b'} &\cdots & \pi_c & \cdots
%\end{ytableau}\,
%\raisebox{1.5mm}{.}\]
%\end{ytableau}.\]
%($\star$ Why does $\pi_b$ necessarily come before $\pi_{b'}$?)
%By our choice of $c$, the path $\pi_c$ lies strictly above $\pi_a$. 
%($\star$ When is this fact used?)
Since $a$, $b'$ belong to no common interval and $a < b'$ by the definition
of $\xi$, we have by Observation~\ref{o:sourcetosink} that there is
no path in $F_w$ from source $a$ to sink $b'$.  Thus $\pi_a$ and $\pi_f$
do not intersect, and  
%Since $a < b'$, the path 
$\pi_f$ must lie entirely above $\pi_a$. 
%Note that $\pi_f$ cannot intersect $\pi_a$, otherwise $b'$ and $a$
%belong to some common interval $[c_i,d_i]$. 
%($\star$ Why is this true, and why is this a problem?)
%But then $\pi_f$ lies above $\pi_a$, 
It follows that $\varphi((\pi_a,\pi_b))=(a,f) \neq (a,c)$, contradiction.
Now suppose that $\pi_a$ does not intersect $\pi_b$, 
and that $a$, $b$ belong to some common interval $[c_i, d_i]$ 
in (\ref{eq:concat2}). 
Let 
$\xi((a,b)) = (\pi_a, \pi_c)$
and let 
$\varphi((\pi_a, \pi_c)) = (a,b')$.
Since $a$ and $c$ belong to no common interval in (\ref{eq:concat2}),
we have $c \neq b'$. 
%($\star$ or what is the reason for mentioning that $a$ and $c$ belong to
%no common interval?)
%By our choice of $\pi_c$, the indices $a$ and $c$ do not belong to any 
%common interval in (\ref{eq:concat2}).  
%We also have $c > a$. ($\star$ When is this used?)
Suppose $b' \neq b$ and let $g$ be the sink index of $\pi_{b'}$.
By the definitions of $\varphi$ and $\xi$, and since 
$\pi_b$ lies entirely above $\pi_a$,
the row $U_{p_1}$ (up to cyclic rotation) has the form
%Then (up to cyclic rotation), $U_{p_2}$ has the form
\begin{equation*}
\cdots\
\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/1_c}}\ 
\cdots\
\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/2_bprimeg}}\ 
\cdots\
\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/1_b}}\ . 
\end{equation*}
%\[\ytableausetup{mathmode}
%\begin{ytableau}
%\cdots & \pi_c & \cdots & \pi_{b'} & \pi_{g} &\cdots & \pi_b & \cdots
%\end{ytableau}\,
%\raisebox{1.5mm}{.}\]
%($\star$ Why does $\pi_{b'}$ necessarily come before $\pi_b$?)
By our choice of $\pi_{b'}$ we have $g > a$, and there is no path in $F_w$
from source $a$ to sink $g$. Thus by Observation~\ref{o:sourcetosink} 
we have that $a$ and $g$ belong to no common interval in (\ref{eq:concat2}).
%($\star$ Why?)  By Observation~\ref{o:sourcetosink}, 
%But this implies that
%there is no path in $F_w$ from $a$ to $g$.  
%($\star$ Why is $\pi_{b'}$ intersecting $\pi_a$ a problem?)
But this implies that $\xi((a,b)) = (\pi_a, \pi_g) \neq (\pi_a, \pi_c)$, 
a contradiction.
%
%Assume there exists a pair $(\pi_a, \pi_b) \in U_{p_1} \times U_{p_2}$ 
%with $b > a$.  
%Suppose $(a,b)$ belongs to the set on 
%the right-hand side of (\ref{eq:indexpairs}).  
%If $\pi_a$ intersects $\pi_b$, then $(\pi_b,\pi_a)$ belongs to 
%the left-hand side of (\ref{eq:indexpairs}).  
%If $\pi_a$ does not intersect $\pi_b$, then it is easy to see that
%$\pi_b$ terminates at sink $k$ for some $k > b$, and that no interval
%$[c_i,d_i]$ contains both $k$ and $a$.
%Now read $U_{p_1}$ cyclically from $\pi_b$ to the right, 
%and let $\pi_{b'}$ be the first path which terminates at a sink
%$\ell$ such that $\ell < a$, or $\ell$ and $a$ belong to the same interval
%$[c_i,d_i]$ for some $i$. Such a path exists, since $U$ is cyindrical
%and the path preceding $\pi_b$
%terminates at sink $b$.  The paths $\pi_{b'}$ and $\pi_a$ 
%satisfy $b' > a$ since $\pi_{b'}$ is the first path after $\pi_k$ to terminate
%at a sink in or below the interval $[c_i,d_i]$.
%The paths intersect because ($\star$ why?)
%Thus $(\pi_{b'}, \pi_a)$ is an inversion and belongs to 
%the left-hand side of (\ref{eq:indexpairs}).  
%($\star$ Why is this map from the right-hand side to the left-hand side 
%injective?)  ($\star$ Need to show that the pair $(a,b')$ does not belong to
%the right-hand side of (\ref{eq:indexpairs})?)  
%Now suppose that $(a,b)$ does not belong to the set on 
%the right-hand side of (\ref{eq:indexpairs}).  
%If $\pi_a$ does not intersect $\pi_b$, then $(\pi_b, \pi_a)$ does not belong to
%the left-hand side of (\ref{eq:indexpairs}).  
%If $\pi_a$ intersects $\pi_b$, then 
%the sink $k$ of $\pi_b$ is either less than $a$, or both $a$ and $k$ belong
%to an interval $[c_i,d_i]$ for some $i$.  Read row $p_1$ cyclically
%from {\em right to left} starting at $\pi_b$, and find the first path
%$\pi_{b'}$ such that both $b'$ and $a$ belong to some interval $[c_i,d_i]$,
%but $\pi_{b'}$ does not intersect $\pi_a$.  Such a path exists, since 
%since $U$ is cylindrical and some path terminates at sink $b$.
%The pair $(\pi_b, \pi_a)$ belongs to 
%the left-hand side of (\ref{eq:indexpairs}), but all such inversions
%correspond to those previously counted.  ($\star$ Why?)
%($\star$ Why is this proof valid?)
\end{proof}
%($\star$ Does the above hold if both 
%instances of ``left-anchored'' are removed?)
%It follows that 
Now we can factor the expressions in (\ref{eq:sumxu}) as follows.

%slightly from the set
%need to consider tableau
%Specifically, 
%If $X$ is an ordered set partition of $[n]$, let 
%Now 
%we have the following.
%($\star$ This proposition still isn't quite right: $V$ needs to be $U$?)
\begin{prop}\label{p:multisum}
Fix $w \in \sn$ \avoidingp, 
$\lambda  = (\lambda_1,\dotsc,\lambda_r) \vdash n$,
and a set partition $I = (I_1,\dotsc,I_r) \vdash [n]$ of type $\lambda$.
Let $V = V(F_w,I)$.
%and a tableau $V \in \mathcal C(I,F_w)$.
Then we have
\begin{equation*}
%\label{eq:multisum}
\sum_U 
q^{\inv(U_1 \circ \cdots \circ U_r)} = 
q^{\inv(V_1 \circ \cdots \circ V_r))}
\Big( \sum_{W^{(1)}} q^{\inv(W^{(1)})} \Big) \cdots
\Big( \sum_{W^{(r)}} q^{\inv(W^{(r)})} \Big),
\end{equation*}
where the sums are over
%\begin{equation*}
$U \in \mathcal C(I,F_w)$, 
$W^{(1)} \in \mathcal C(I_1, F_w|_{I_1}), \dotsc,
W^{(r)} \in \mathcal C(I_r, F_w|_{I_r})$,
%\end{equation*}
or over
%\begin{equation*}
$U \in \mathcal C_L(I,F_w)$, 
$W^{(1)} \in \mathcal C_L(I_1, F_w|_{I_1}), \dotsc,
W^{(r)} \in \mathcal C_L(I_r, F_w|_{I_r})$.
%\end{equation*}
%The analogous equation with sums over   
%$\mathcal C_L(I,F_w), 
%\mathcal C_L(I_1, F_w|_{I_1}), \dotsc, \mathcal C_L(I_1, F_w|_{I_1})$
%also holds.
%where the 
%right-hand sum is over
%(left-anchored) 
%cylindrical $F_w$-tableaux $U$ of shape $\lambda$ such that
%A$L(U_j)$ contains the elements $I_j$ for $j = 1,\dotsc,r$,
%$V$ is any cylindrical $F_w$-tableaux of 
%and 
%(left-anchored) 
%cylindrical $F_w|_{I_j}$-tableaux $U^{(j)}$ of shape 
%$(\lambda_j)$, for $j = 1,\dotsc, r$.
\end{prop}
\begin{proof}
First observe that we have a bijection
$\mathcal C(I_1, F_w|_{I_1}) \times \cdots \times \mathcal C(I_r, F_w|_{I_r})
\rightarrow \mathcal C(I,F_w)$ defined by joining $W^{(1)}, \dotsc, W^{(r)}$
into a single tableau $U$ with $U_j = W^{(j)}$.
Clearly, the bijection restricts to the 
corresponding subsets of left-anchored tableaux, and satisfies
\begin{equation}\label{eq:multisum}
\inv(U_1 \circ \cdots \circ U_r) = 
\inv(W^{(1)}) + \cdots + \inv(W^{(r)})
+ \inv(U^\tr).
%\inv((U_1)^\tr \circ \cdots \circ (U_r)^{\tr}).
\end{equation}
Now observe that the number $\inv(U^\tr)$
%$\inv((U_r)^\tr \circ \cdots \circ (U_1)^{\tr})$
is equal to the left-hand side of (\ref{eq:indexpairs}), 
summed over pairs $(p_1, p_2)$ with $p_1 < p_2$.
Furthermore, by Observation~\ref{o:sourcetosink}, Lemma~\ref{l:dsnintersect}, 
and the definition of $V(F_w,I)$,
we have that $\inv(V_1 \circ \cdots \circ V_r)$ 
is equal to the right-hand side of (\ref{eq:indexpairs}),
summed over pairs $(p_1, p_2)$ with $p_1 < p_2$.
Thus we may rewrite (\ref{eq:multisum}) as
%by Lemma~\ref{l:indexpairs} we have that the above bijection satisfies
\begin{equation*}
\inv(U_1 \circ \cdots \circ U_r) = 
\inv(W^{(1)}) + \cdots + \inv(W^{(r)})
+ \inv(V_1 \circ \cdots \circ V_r),
\end{equation*}
as desired.
%Thus by Observation~\ref{o:pinvdecomp}, Equation (\ref{eq:multisum}) 
%becomes true if $V_r, \dotsc, V_1$ are replaced by $U_r,\dotsc, U_1$, 
%respectively.  Now consider the exponent 
%%$\inv((V_r)^\tr \circ \cdots \circ (V_1)^\tr)$.
%$\inv(V_r \circ \cdots \circ V_1)$.
%By the definition of $V(F_w,I)$, this is equal to the right-hand side of 
%(\ref{eq:indexpairs}). ($\star$ What did I mean to write?  The cardinality of
%the union of right-hand sides over all pairs of rows?)
%On the other hand, the left-hand side of (\ref{eq:indexpairs}) is equal
%to $\inv((U_r)^\tr \circ \cdots \circ (U_1)^\tr)$.
%($\star$ What did I mean to write here?)
\end{proof}



%We claim that for $U,V \in \mathcal C(I,F_w)$, we have
%\begin{equation}\label{eq:strongerinveq}
%\inv(U_r \circ \cdots \circ U_1) =
%\inv((V_r)^\tr \circ \cdots \circ (V_1)^\tr)
%+ \inv(U_r) + \cdots + \inv(U_1).
%\end{equation}
%Fix $U \in \mathcal C(I,F_w)$, let 

%\begin{proof}
%Now observe that since for any two tableaux $U, V \in \mathcal C(I,F_w)$,
%the corresponding rows are equal as sets,
%we have equality of columns as sets 
%in the reverse transpose tableaux, and thus 
%\begin{equation*}
%\inv((U_r)^\tr \circ \cdots \circ (U_1)^\tr) = 
%\inv((V_r)^\tr \circ \cdots \circ (V_1)^\tr).
%\end{equation*}
%This proves (\ref{eq:strongerinveq}), from which (\ref{eq:multisum}) follows.
%\end{proof}


It is clear that a cylindrical tableau $U$ is completely determined
by the Young tableau $L(U)$.  Under some conditions the insertion
or deletion of a greatest letter in the left tableau of a cylindrical
tableau yields a valid left tableau of another cylindrical tableau.
In these cases, intersecting paths in the two cylindrical tableaux
are closely related.

%Thus we have 
%\begin{equation}\label{eq:CY}
%\mathcal C([n],F_w) = \bigcup_{i=1}^n Y_n(F_w,i),
%\qquad 
%\mathcal C_L([n],F_w) = Y_n(F_w,1).
%\end{equation}

\begin{lem}\label{l:ndelins}
Let $F_w$ be a connected 
zig-zag
%descending star 
network of order $n$
corresponding to the concatenation (\ref{eq:concat2}) of star networks
%Let indices $t$, $c_1,\dotsc,c_t$, $d_1,\dotsc,d_t$, be as in 
(\ref{eq:concat2}),
and let $[b,n]$ be the unique interval containing $n$.
%where $I$ is an ordered 
\begin{enumerate}
\item For any cylindrical $F_w$-tableau $U$ of shape $(n)$,
if $T'$ is the Young tableau obtained from $L(U)$ by deleting
the entry $n$, then there exists a unique 
cylindrical $F_{\iota(w)}$-tableau $U'$
such that $T' = L(U')$.
\item For any cylindrical $F_{\iota(w)}$-tableau $V'$ of shape $(n-1)$,
let $T$ be the Young tableau obtained from $L(V')$ 
by inserting the entry $n$ cyclically before any element in $[b, n-1]$ 
if the interval $[b,n]$ is maximal in $\preceq$, 
and cyclically after any element in $[b,n-1]$ otherwise.
Then there exists a unique
cylindrical $F_w$-tableau $V$ such that $L(V) = T$.
\item Let tableaux $U$ \ntnsp and $U'$ in (1) contain the path families
$(\pi_1, \dotsc, \pi_n)$ and $(\pi'_1, \dotsc, \pi'_{n-1})$, respectively.
For all pairs $(i,j)$, if $\pi_i$, $\pi'_i$ have the same sink
index, and $\pi_j$, $\pi'_j$ have the same sink index,
then $\pi_i$ and $\pi_j$ intersect 
if and only if $\pi'_i$ and $\pi'_j$ intersect.
\end{enumerate}
\end{lem}
\begin{proof}
%It is clear that for any zig-zag network $F$ and any Young tableau $T$, 
%if there is any cylindrical $F$-tableau $U$ satisfying $L(U) = T$,
%then this tableau is unique. 
%Since $F_w$ is connected, we have $d_1 = n$.
%\noindent
(1) Let $(i_1, i_2)$ be a pair of cyclically consecutive entries in $T'$.
If these entries are also cyclically consecutive in $L(U)$, then there exists
a (unique) 
path $\pi_{i_1}$ in $F_w$ from source $i_1$ to sink $i_2$.  
Since $F_{\iota(w)}$ differs from $F_w$ by the removal of the 
unique path from source $n$ to sink $n$
and possibly the contraction of an edge to a single vertex, 
the image of $\pi_{i_1}$ is again the unique path in $F_{\iota(w)}$ from
source $i_1$ to sink $i_2$.
If $(i_1, i_2)$ are not cyclically consecutive in $L(U)$, then $i_1$
cyclically precedes $n$ 
and $i_2$ cyclically follows $n$ 
%and $i_2$ cyclically follows $n$ 
in $L(U)$.
Thus either $i_1$ or $i_2$ belongs to $[b,n-1]$.
Thus in $G$ there is a path from source $i_1$ to the central vertex
of $G_{[b,n]}$ and a path from this vertex to sink $i_2$.
It follows that there is a path in $F_w$ from source $i_1$ to sink $i_2$.
%, and therefore belongs to the interval $[c_1, n-1]$.
%Since $F_w$ is a connected descending star network, the partial order $\prec$
%is a chain and by Observation~\ref{o:sourcetosink}, there is a path 
%from source $i_1$ to every sink in $F_{\iota(w)}$.
Uniqueness of $U'$ follows from uniqueness of source-to-sink paths
in zig-zag networks.  (See comment following Theorem~\ref{t:lem53nndcb}.)

%\noindent 
(2)
%By Proposition~\ref{p:iotainverse}, 
%the descending star network 
%$F_w$ is uniquely determined by the pair $(F_{\iota(w)}, \ell = c_1)$,
%and corresponds to the concatenation (\ref{eq:concat2}) with $d_1 = n$.
Let $(i_1, i_2)$ be a pair of cyclically consecutive entries in $T$.
If these entries are also cyclically consecutive in $L(V')$,
then there exists a (unique) 
path from source $i_1$ to sink $i_2$ 
in $F_{\iota(w)}$.  Since each interval in the concatenation 
corresponding to $F_{\iota(w)}$ is equal to or is contained in an interval
in the concatenation corresponding to $F_w$, we use
Observation~\ref{o:sourcetosink} and the fact that both $F_w$ and $F_{\iota(w)}$
are connected to infer that there is a path in $F_w$ from source $i_1$ to 
sink $i_2$. 
Again, uniqueness of $V$ follows from uniqueness of 
source-to-sink paths in zig-zag networks.  

%\noindent 
(3)
Let $\pi_i$, $\pi'_i$, $\pi_j$, $\pi'_j$ satisfy the stated conditions.
Then the source and sink indices of these paths are not equal to $n$.
Suppose first that in the concatenation (\ref{eq:concat2})
corresponding to $F_w$ we have $t=1$, or $d_1 = n$ and $d_2 < n-1$, 
or $d_t = n$ and $d_{t-1} < n-1$.
Then $F_{\iota(w)}$ is the subgraph of $F_w$ obtained by 
deleting the unique path from source $n$ to sink $n$.
By the uniqueness of paths in descending star networks, we have
$\pi_i = \pi'_i$ and $\pi_j = \pi'_j$.
Now suppose that in (\ref{eq:concat2}) we have 
$d_1 = n$ and $d_2 = n-1$, 
or $d_t = n$ and $d_{t-1} = n-1$, 
and let $x$, $y$ be the vertices in $F_w$ corresponding to
the central vertices of the star networks 
$G_{[b,n]}$ and $G_{[c_2,d_2]}$, respectively,
or $G_{[c_{t-1},d_{t-1}]}$ and $G_{[b,n]}$, respectively.
Then $F_{\iota(w)}$ is obtained from $F_w$ by
deleting the unique path from source $n$ to sink $n$,
and by contracting the edge $(x,y)$ to a single vertex $z$.
Thus if $\pi_i \cap \pi_j$ contains the edge $(x,y)$
then $\pi'_i \cap \pi'_j$ contains the vertex $z$;
if $\pi_i \cap \pi_j$ does not contain the edge $(x,y)$,
%is empty in $F_w$,
then $\pi'_i \cap \pi'_j = \pi_i \cap \pi_j$. 
%is empty in $F_{\iota(w)}$.
%intersect in $F_w$,
%Then either $\pi_i = \pi'_i$, or 
%$\pi'_i$ obtained by contracting
%and $\pi'_i$ differ at most
%and $\pi_j = \pi'_j$,
%Suppose $i < j$, and let $m$ be the sink of both $\pi_j$ and $\pi'_j$.
%Then by Lemma~\ref{l:zzintersect}, 
%$\pi_i$ intersects $\pi_j$ if and only if there exists a path
%in $F_w$ from $i$ to $m$.
%By Observation~\ref{o:sourcetosink}, and
%since $F_w$ is connected, such a path exists if and only if 
%either $m < i$ or $i$ and $m$ belong to a common interval
%in the concatenation (\ref{eq:concat2}) corresponding to $F_w$.
%To see that this map is well-defined, note that 
%The deletion of $n$ from $L(U)$ results in a valid left tableau
%of an element of $Y_{n-1}(F_{\iota(w)},i)$, i.e., for each pair of consecutive
%entries $(i_1, i_2)$ in this tableau, there is a unique path from source 
%$i_1$ to sink $i_2$ in $F_{\iota(w)}$. 
%This is the image of a path in $F_w$ unless $i_1$ precedes $n$ and $i_2$
%follows $n$ in $U$.  In this case however, 
%corresponds to a path
%in $F_{\iota(w)}$ unless $i_1 = n$ or $i_2 = n$.
%the index preceding 
%$n$ in $L(U)$ 
%$i_1$ necessarily belongs to the interval $[c_1,n-1]$
%$i_{\ell-1}$ necessarily
%belongs to the interval $[c_1,n-1]$ %
%and from each source in this range, there exists a unique path 
%in $F_{\iota(w)}$ to $i_2$, in fact to each sink $1, \dotsc, n-1$.
\end{proof}

%($\star$ Give an example.)

%($\star$ Introduce this.)

Given a zig-zag network $F$ of order $n$, define $Y_n(F,i)$ 
to be the set of 
%connected 
cylindrical $F$-tableaux $U$ of shape $(n)$ 
in which the first entry of $L(U)$ is $i$.
In terms of our earlier notation, we have
\begin{equation}\label{eq:CY}
\mathcal C([n], F_w) = \bigcup_{1 \leq i \leq n} Y_n(F_w, i),
\qquad
\mathcal C_L([n], F_w) = Y_n(F_w, 1),
\end{equation}
where we interpret $[n]$ as the ordered set partition having one block.

Let us examine the map 
$U \mapsto U'$ and the numbers $\inv(U)$, $\inv(U')$
defined by Lemma~\ref{l:ndelins} (1) in the case that
$U \in Y(F_w, i)$ and $i \leq n-1$.
To be precise,
%($\star$ Introduce the maps $\delta_1$, $\delta_2$ too.)
for each pair $(F_w, i)$ where $F_w$ 
is a connected zig-zag network of order at least 2
with corresponding concatenation (\ref{eq:concat2}),
poset $\preceq$ of intervals,
and $[b,n]$ the unique interval containing $n$,
%$w \in \sn$ avoiding the pattern $312$ 
and $1 \leq i \leq n-1$, we define a map 
\begin{equation*}
%\label{eq:delta1def}
\begin{aligned}
\delta_1: Y_n(F_w, i) &\rightarrow
Y_{n-1}(F_{\iota(w)}, i) \times \{ 0, 1, \dotsc, n - b - 1 \}\\
U &\mapsto (U',k)
\end{aligned}
\end{equation*}
by declaring $U'$ to be the cylindrical tableau 
whose left tableau is obtained from $L(U)$
% = (i_1, \dotsc, i_n)$ 
by deleting $n$,
%$i_\ell = n$, 
and by declaring 
$k$ to be the number of paths following $\pi_n$ in $U$ whose
\begin{equation*}
\begin{cases}
\text{sink index belongs to $[b,n-1]$} 
&\text{if $[b,n]$ maximal in $\preceq$},\\
\text{source index belongs to $[b,n-1]$} 
&\text{otherwise}.
\end{cases}
\end{equation*}
%in $\{\pi_{i_{\ell+1}}, \dotsc
%$U$ forming an inversion with $\pi_n$.
% of the form $(\pi_n, \pi_j)$ in $U$.
%indices in the interval $[b, n-1]$ 
%%such that $\pi_i$ appears to the right of $\pi_n$ in $U$.
%appearing to the right of $n$ in $L(U)$.
%\begin{equation*}
%k = \begin{cases}
%\# \{ j \in [b,n-1] \,|\, b \text{appears to the right of $n$ in $L(U)$} \},\\
%\# \{ j \in [b,n-1] \,|\, b \text{appears to the right of $n$ in $R(U)$} \}-1.
%\end{cases}
%\end{equation*}
%By Lemma~\ref{l:ndelins} and the fact that $i \leq n-1$,
%this map is well-defined. 
%= (j, i_2, \dotsc, i_n)$ with $i_\ell = n$ as follows.
%\begin{equation*}
%$U = (\pi_j, \pi_{i_2}, \dotsc, \pi_{i_n})$ with $i_\ell = n$ as follows.
%\begin{enumerate}
%\item Declare $k$ to be the number of indices $i \geq c_1$ 
%such that $\pi_i$ appears to the right of $\pi_n$ in $U$.
%appearing to the right of $n$ in $U$.
%\item For $r \not \in \{ \ell-1, \ell\}$, replace the path $\pi_{i_r}$ in $F_w$ 
%from source $i_r$ to sink $i_s$ 
%by the unique path in $F_{\iota(w)}$ 
%from source $i_r$ to sink $i_s$. 
%\item Delete $n$.
%Replace $\pi_{\ell-1}, \pi_{i_\ell}$
%by the unique path in $F_{\iota(w)}$ 
%from source $i_{\ell - 1}$ to the sink of $\pi_n$.
%\end{enumerate}

For example, let $n = 6$, recall 
%the descending star network $F_{256431}$ in 
(\ref{eq:F256431}), and consider the $F_{256431}$-tableau and its left 
Young tableau
\begin{equation*}
U = 
\raisebox{-1.6mm}{\includegraphics[height=6mm]{tableaux/6_466112255334}}\
%\in Y_6(256431,4), \quad
, \quad
T =
\raisebox{-1.6mm}{\includegraphics[height=6mm]{tableaux/6_461253}}\ ,
\end{equation*}
with $U \in Y_6(256431,4)$, and
$[3,6]$ not maximal in the poset 
$[3,6] \prec [2,4] \prec [1,2]$.
Removing $6$ from $T$ we have the tableaux
\begin{equation*}
T' =
\raisebox{-1.6mm}{\includegraphics[height=6mm]{tableaux/5_41253}}\ , \quad
U' = 
\raisebox{-1.6mm}{\includegraphics[height=6mm]{tableaux/5_4112255334}}\ ,
\end{equation*}
with
$T' = L(U')$ and
$U' \in Y_5(F_{25431},4)$.
Since 
%$[3,6]$ is not maximal in the poset 
%$[3,6] \prec [2,4] \prec [1,2]$,
the only paths in $U$ which follow $\pi_6$ and have source indices
in $[3,5]$ are $\pi_5$, $\pi_3$, 
%in $U$ forming inversions with $\pi_6$ are 
%$\pi_4$ and $\pi_3$, 
we have $\delta_1(U) = (U', 2)$.

\begin{lem}\label{l:delta1}
For each connected zig-zag 
network $F_w$ of order $n$, 
the map $\delta_1$ 
%(\ref{eq:delta1def}) 
is a bijection.  
Furthermore, if $\delta_1(U) = (U',k)$ then we have $\inv(U) = \inv(U') + k$.
\end{lem}
\begin{proof}
Assume that $F_w$ 
%is a connected descending star network of order $n$,
corresponds to the concatenation (\ref{eq:concat2})
in which the unique interval containing $n$ is $[b,n]$,  
%($\star$ Now how is $c_1$ defined?)
%To invert $\delta_1$, 
and let $(U',k)$ be a pair in 
$Y_{n-1}(F_{\iota(w)}, i) \times \{ 0, 1, \dotsc, n - b - 1 \}$.
%and assume that $F_{\iota(w)}$ corresponds to the concatenation
%($\star$ $v = \iota(w)$.  Do we have to select $w$ first?)
%$w \in \sn$ to be 
%the $312$-avoiding permutation 
%whose descending star network 
To invert $\delta_1$, find an entry $j \in [b,n-1]$ in the tableau $L(U')$
with exactly $k-1$ entries in $[b,n-1]$ to its right.  
(This is possible, since $k \leq n-1-b$.)
Now create a new Young tableau $T$ by inserting the letter $n$ into $L(U')$ 
immediately to the left of $j$ if $[b,n]$ is maximal in $\preceq$, 
or immediately to the right of $j$ otherwise.
By Lemma~\ref{l:ndelins} (2)
%We claim that 
there is a unique $F_w$-tableau $U$ with $L(U) = T$.
%Indeed, for consecutive indices $(i_1, i_2)$ in $U''$ with $i_1, i_2 \neq n$,
%there is a unique path from source $i_1$ to sink $i_2$ in $F_w$.
%If $i_2 = n$ then $i_1 = j \in [c_1,n-1]$ and there is a unique
%path from $j$ to $n$ in $F_w$.
%If $i_1 = n$, then there is a unique path from source $n$ to sink $i_2$,
%indeed to any sink.

%By Proposition~\ref{p:iotainverse}, there is a unique
%descending star network $F_w$ with ($\star$ some property?).
%Furthermore, the Young tableau $U''$ defines a 
%unique cylindrical $F_w$-tableau $U$ satisfying $L(U) = U''$.
%($\star$ Substantial work needed on this proof.)
%By the position of $n$ in $U''$, 

To compare inversions in $U$ 
%
and $U'$, write
\begin{equation*}
\begin{aligned}
U &= (\pi_{i_1}, \dotsc, \pi_{i_n}),\\
% \text{ with } \pi_{i_\ell} = \pi_n,\\ 
%\pi_{i_{\ell+1}}, \dotsc, \pi_{i_n}),\\
%U  = (\pi_{i_1}, \dotsc, \pi_{i_{\ell-1}}, \pi_{i_\ell} = \pi_n, 
%\pi_{i_{\ell+1}}, \dotsc, \pi_{i_n}),\\
U' &= (\pi'_{i_1}, \dotsc, \pi'_{i_{\ell-1}}, \pi'_{i_{\ell+1}}, \dotsc, \pi'_{i_n}),
\end{aligned}
\end{equation*}
where $\pi_{i_\ell} = \pi_n$.
Observe that all paths except $\pi_{i_{\ell-1}}$, $\pi_{i_\ell}$
% = \pi_n$
%$\pi_{i_1}, \dotsc, \pi_{i_{\ell-2}}, \pi_{i_{\ell+1}}, \dotsc, \pi_{i_n}$ 
in $U$ have the same sources and sinks as the corresponding paths in $U'$.
%\begin{equation*}
%U' = 
%\begin{cases}
%(\pi'_{i_2}, \dotsc, \pi'_{i_n}) &\text{if $[b,n]$ is maximal in $\preceq$},\\
%(\pi'_{i_n} = \pi'_j, \pi'_{i_2}, \dotsc, \pi'_{i_{n-1}}) &\text{otherwise}.
%\end{cases}
%\end{equation*}
Thus by Lemma~\ref{l:ndelins} (3), two such paths form an inversion
in $U$ if and only if the corresponding paths form an inversion in $U'$.
Consider therefore inversions in $U$ which involve one of the paths
$\pi_{i_{\ell-1}}$, $\pi_{i_\ell}$,
% = \pi_n$,
%or $\pi_{i_n}$, 
and inversions in $U'$ which involve the path $\pi'_{i_{\ell-1}}$.

Suppose first that $[b,n]$ is maximal in $\preceq$.
By definition, there are $k$ inversions in $U$ of the form 
$(\pi_{i_\ell}, \pi_c)$, where $\pi_c$ terminates at a sink 
having index in $[b,n-1]$.  These in fact are the only inversions in $U$
involving $\pi_{i_\ell}$: since there are no paths in $F_w$ from source $n$
to sinks $1, \dotsc, b-1$, Observation~\ref{o:pnetintersect}
implies that $\pi_{i_\ell}$ cannot intersect any path terminating at one
of these sinks.
%For each path $\pi_c$ terminating 
%at another sink, we claim that $(\pi_n,\pi_c)$ is an inversion in $U$
%if and only if $(\pi'_{i_{\ell-1}}, \pi'_c)$ is an inversion in $U'$. 
%and let $U' = (\pi'_{i_1}, \dotsc, \pi'_{i_n})$.
%Then paths 
Now observe that path $\pi_{i_{\ell-1}}$ terminates at sink $n$,
while paths $\pi'_{i_{\ell-1}}$ and $\pi_n$ terminate at sink $i_{\ell+1}$.
%If $[b,n]$ is maximal in $\preceq$, then 
By the maximality of $[b,n]$, we have that $i_{\ell+1}$ 
belongs to the interval $[b,n]$. 
%Since $\pi_n$ also terminates at sink $i_{\ell+1}$.
Thus the paths $\pi_{i_{\ell-1}}$ and $\pi'_{i_{\ell-1}}$ are identical
from their sources 
%$i_{\ell_1}$ in $F_w$ ($F_{\iota(w)}$) 
up to the vertex of $F_w$ ($F_{\iota(w)}$) 
corresponding to the central vertex of $G_{[b,n]}$.
It follows that any path in $U$ intersects $\pi_{i_{\ell-1}}$ if and only if the
corresponding path in $U$ intersects $\pi'_{i_{\ell-1}}$.
Therefore we have $\inv(U) = \inv(U') + k$.

Now suppose that $[b,n]$ is not maximal in $\preceq$. 
Then it must be minimal, 
and since $\pi_{i_{\ell-1}}$ terminates at sink $n$,
we have $i_{\ell-1} \geq b$.
% belongs to the interval $[b,n-1]$. 
Consider paths $\pi_c$ with $c \geq b$.
%It follows that for any index $c \geq b$, 
%each pair of the form 
By the minimality of $[b,n]$ in $\preceq$,
we have that $\pi_c$ intersects $\pi_{i_{\ell-1}}$ and $\pi_{i_\ell}$
at the vertex of $F_w$ ($F_{\iota(w)}$) corresponding to the
central vertex of $G_{[b,n]}$.
Thus
$(\pi_c, \pi_{i_{\ell-1}})$ 
or
$(\pi_{i_{\ell-1}}, \pi_c)$ 
is an inversion in $U$ if and only if the corresponding pair
is an inversion in $U'$.
By definition, there are $k$ inversions in $U$ 
of the form $(\pi_{i_\ell}, \pi_c)$ with $c \geq b$.
Now consider paths $\pi_c$ with $c < b$.
By Observation~\ref{o:pnetintersect},
no pair $(\pi_{i_{\ell-1}}, \pi_c)$ 
is an inversion in $U$,
since there is no path from source $c$ to sink $n$ in $F_w$
On the other hand, the paths $\pi_{i_\ell}$ and $\pi'_{i_{\ell-1}}$
%both terminate at sink $i_{\ell+1}$ and 
are identical from the vertex of $F_w$ (or $F_{\iota(w)}$) corresonding
to the central vertex of $G_{[b,n]}$ to sink $i_{\ell+1}$.
Thus each pair $(\pi_{i_\ell}, \pi_c)$ is an inversion in $U$
if and only if $(\pi'_{i_{\ell-1}}, \pi'_c)$ is an inversion in $U'$.
It follows again that $\inv(U) = \inv(U') + k$.
%To see that the tableau $U$ satisfies $\inv(U) = \inv(U') + k$, 
%let $(\pi_1, \dotsc, \pi_n)$ be the path family contained in $U$,
%and let $(\pi'_1, \dotsc, \pi'_{n-1})$ be the path family contained in $U'$.
%Consider subsequences $(\pi_{\ell_2}, \pi_{\ell_1})$
%and $(\pi'_{\ell_2}, \pi'_{\ell_1})$ of paths in $U$ and $U'$, with 
%$\ell_1 < \ell_2 \leq n$.
%Suppose first that we have $\ell_1 \geq c_1$.
%If $\ell_2 \leq n-1$, then both subsequences are inversions,
%since any two paths starting at sources with indices in the interval
%$[c_1,n-1]$ necessarily intersect.
%If $\ell_2 = n$, then $(\pi_{\ell_2}, \pi_{\ell_1})$ is an inversion
%for the same reason, and there are $k$ such inversions by our choice of $j$.
%There is no path $\pi'_n$.
%Now suppose that we have $\ell_1 < c_1$.
%Then $\ell_1 \neq j \geq c_1$. 
%If $\ell_2 \not \in \{j, n\}$,
%then the source and sink indices of the paths $(\pi_{\ell_2}, \pi_{\ell_1})$ 
%are the same as those of the paths $(\pi'_{\ell_2}, \pi'_{\ell_1})$.
%Thus by Lemma~\ref{l:ndelins} (3),
%the pair $(\pi_{\ell_2}, \pi_{\ell_1})$ 
%is an inversion in $U$ 
%if and only if 
%the pair $(\pi'_{\ell_2}, \pi'_{\ell_1})$ is an inversion in $U'$.
%If $\ell_2 = j$, then $\pi_{\ell_2}$ terminates at sink $n$.  Since 
%there is no path in $F_w$ from source $\ell_1 < c_1$ to sink $n$, the paths
%$\pi_{\ell_1}$ and $\pi_{\ell_2}$ do not intersect, 
%by Lemma~\ref{l:dsnintersect}.
%Thus $(\pi_j, \pi_{\ell_1})$ is not an inversion in $U$, but
%$(\pi'_j, \pi'_{\ell_1})$ may be an inversion in $U'$.
%If $\ell_2 = n$, then the path $\pi'_{\ell_2}$ does not exist, but
%$(\pi_n, \pi_{\ell_1})$ may be an inversion in $U$.
%We claim that $(\pi_n, \pi_{\ell_1})$ is an inversion in $U$ if and only if
%$(\pi'_j, \pi'_{\ell_1})$ is an inversion in $U'$.
%Let $m$ be the sink of $\pi_n$, which by the definition of $\delta_1$ is
%also the sink of $\pi'_j$.
%By Lemma~\ref{l:dsnintersect}, 
%the pair $(\pi_n, \pi_{\ell_1})$ is an inversion in $U$ if and only
%if there is a path in $F_w$ from $\ell_1$ to $m$.
%Since $\ell_1 < c_1$, 
%such a path in $F_w$ exists if and only if a path from $\ell_1$ to $m$
%exists in $F_{\iota(w)}$.  By Lemma~\ref{l:dsnintersect} again,
%such a path in $F_{\iota(w)}$ exists if and only if 
%%there is a path from $\ell_1$ to $m$ in $F_{\iota(w)}$.
%$(\pi'_j, \pi'_{\ell_1})$ is an inversion in $U'$.
\end{proof}

Note that in the example preceding Lemma~\ref{l:delta1}
we have $\inv(U) = 6 = \inv(U') + 2$, and
$\delta_1(U) = (U',2)$.
%%the unique interval containing $m = 6$ in 
%the $F_{246531}$-tableau $U$ has two paths which form inversions with $\pi_6$:
%$\pi_{99}$ (\ref{eq:F246531}) 
%%of order $m = 6$ 
%corresponds to the concatenation $G_{[3,6]} \circ G_{[2,4]} \circ G_{[1,2]}$,
%and the interval $[3,6]$ is not maximal in the poset 
%$[3,6] \prec [2,4] \prec [1,2]$.
%The tableaux $U$, $U'$ satisfy
%$j = 4$, $\inv(U) = 6$, $\inv(U') = 3$, and
%$\inv(U) = \inv(U') + 2m - 2j - 1$.
%($\star$ Later?)
%, and $\inv(U') = 4$.
%In the concatenation corresponding to $F_{256431}$, we have $b = 3$,
%and in $T$, the number of entries in $[3,5]$ to the right of $6$ is $2$.
%The tableaux $U$ and $U'$ satisfy
%%\begin{equation*}
%$\inv(U) = 6 = \inv(U') + 2$.
%%\end{equation*}

%If $\ell_1 \neq j$, $\ell_2 \not \in \{j, n\}$,
%%$j \not \in \{ \ell_1, \ell_2 \}$, 
%then the pair $(\pi_{\ell_2}, \pi_{\ell_1})$ is an inversion in $U$ 
%if and only if 
%the pair $(\pi'_{\ell_2}, \pi'_{\ell_1})$ is an inversion in $U'$
%by Lemma~\ref{l:ndelins} (3).
%If $\ell_1 = j$ or $\ell_2 = j$
%
%If $\ell_1 = j$, then we have $c_1 \leq \ell_1 < \ell_2 \leq n-1$,
%so
%$(\pi_{\ell_2}, \pi_{\ell_1})$ is an inversion in $U$,
%and  
%$(\pi'_{\ell_2}, \pi'_{\ell_1})$ is an inversion in $U'$.
%($\star$ Why?)
%%Now consider subsequences
%%$(\pi_{\ell_2}, \pi_{\ell_1})$ with $\ell_2 \in \{ j, n \}$,
%%$(\pi'_{\ell_2}, \pi'_{\ell_1})$ with $\ell_2 = j$.
%If $\ell_2 = n$, and $\ell_1 \geq c_1$, then 
%$(\pi_{\ell_2}, \pi_{\ell_1})$ is an inversion in $U$,
%and there are $k$ such inversions.
%($\star$ Why?)
%No path $\pi'_n$ exists.
%%($\star$ What about if $\ell_2 = n$ and $c_1 > \ell_1$?)
%If $\ell_2 = j$ and $\ell_1 \geq c_1$, then
%$(\pi_{\ell_2}, \pi_{\ell_1})$ is an inversion in $U$,
%and  
%$(\pi'_{\ell_2}, \pi'_{\ell_1})$ is an inversion in $U'$.
%($\star$ Why?)
%
%Now suppose that we have $\ell_1 < c_1$.
%We claim that $(\pi_n, \pi_{\ell_1})$ is an inversion in $U$ if and only if
%$(\pi'_j, \pi'_{\ell_1})$ is an inversion in $U'$.
%Let $m$ be the sink of $\pi_n$, which by the definition of $\delta_1$ is
%also the sink of $\pi'_j$.
%By Lemma~\ref{l:zzintersect}, 
%the pair $(\pi_n, \pi_{\ell_1})$ is an inversion in $U$ if and only
%if there is a path in $F_w$ from $\ell_1$ to $m$.
%Since $\ell_1 < c_1$, 
%%there is a path from $\ell_1$ to $m$ in $F_w$
%such a path in $F_w$ exists if and only if a path from $\ell_1$ to $m$
%exists in $F_{\iota(w)}$.  By Lemma~\ref{l:zzintersect} again,
%such a path in $F_{\iota(w)}$ exists if and only if 
%%there is a path from $\ell_1$ to $m$ in $F_{\iota(w)}$.
%$(\pi'_j, \pi'_{\ell_1})$ is an inversion in $U'$.
%\end{proof}
%

Now let us examine a map 
$U \mapsto U'$ and the numbers $\inv(U)$, $\inv(U')$
closely related to those defined by Lemma~\ref{l:ndelins} (1) 
in the case that
$U \in Y(F_w, n)$.
% and $i \leq n-1$.
To be precise,
for each connected zig-zag
%descending star 
network $F_w$ of order at least $2$
with corresponding concatenation (\ref{eq:concat2}), poset $\preceq$
of intervals, 
%of $t$ star networks
and $[b,n]$ the unique interval containing $n$,
%$w \in \sn$ avoiding the pattern $312$ 
%and $1 \leq j \leq n-1$, 
we define a map 
\begin{equation}\label{eq:delta2}
\begin{aligned}
\delta_2: Y_n(F_w, n) &\rightarrow
\bigcup_{j=b}^{n-1} Y_{n-1}(F_{\iota(w)}, j),\\
U &\mapsto U',
%(U',k),
\end{aligned}
\end{equation}
by declaring $U'$ to be the cylindrical tableau 
%where $U' = (\pi_{i_n}, \pi_{i_2}, \dotsc, \pi_{i_{n-1}} )$
%is the tableau 
whose left tableau 
%$L(U') = (i_n, i_2, \dotsc, i_{n-1} )$
is obtained from $L(U) = (i_1, \dotsc, i_n)$
%\begin{equation*}
%$U = (\pi_{i_1}, \dotsc, \pi_{i_n})$ by deleting $\pi_{i_1} = \pi_n$
by 
\begin{equation*}
L(U') = 
\begin{cases}
(i_2, \dotsc, i_n) &\text{if $[b,n]$ maximal in $\preceq$},\\
(i_n, i_2, \dotsc, i_{n-1}) &\text{otherwise}.
%&\text{if $[b,n]$ minimal in $\preceq$ and $t>1$},\\
\end{cases}
\end{equation*}
%deleting $i_1 = n$
%and moving $i_n$ to the first position.
%$U' = (\pi_{i_n}, \pi_{i_2}, \dotsc, \pi_{i_{n-1}} )$,
%and by declaring $k = i_n$.
The tableau $U'$ exists and is unique by Lemma~\ref{l:ndelins} (1).
%($\star$ What if $t = 1$, i.e., $[b,n]$ is both maximal and minimal?)
%To see that this map is well-defined, note that 
%there is a path in $G$ from source $i_n$ to the central vertex of $G_{[b,n]}$
%since $i_n$ precedes $n$ cyclically in $U$.  Similarly,
%there is a path in $G$ from the central vertex of $G_{[b,n]}$ to
%sink $i_2$.
%It follows that 
If $[b,n]$ is maximal in $\preceq$ then any path beginning at source $n$
must terminate at a sink in this interval.
Thus we have $i_2 \in [b,n-1]$.
If $[b,n]$ is not maximal in $\preceq$ it must be minimal, 
and any path terminating at sink $n$
must begin at a source in this interval.
Thus we have $i_n \in [b,n-1]$.
%we have $i_n \in [b,n-1]$;
It follows that $U'$ belongs to the union in (\ref{eq:delta2}).

%$i_n$ belongs to the
%interval $[c_1,n-1]$ since it precedes $n$ cyclically.
%Thus $(i_n, i_1, \dotsc, i_{n-1})$ is a valid left tableau
%of an element of the set $Y_n( F_w, j)$ for an index $j$
%in the interval $[c_1, n-1]$.
%By Lemma~\ref{l:ndelins} (1)
%%the argument following the definition of $\delta_1$,
%we may delete $n$ to obtain a valid left tableau
%of an element of $Y_{n-1}(F_{\iota(w)},j)$.
%% \cup \cdots \cup Y_{n-1}(F_{\iota(w)},n-1).$
%$\star$ Give an example.
For example, let $n = 6$, recall 
%the descending star network $F_{256431}$ in 
(\ref{eq:F246531}), and consider the $F_{246531}$-tableau and its left 
Young tableau
\begin{equation*}
U = 
\raisebox{-1.6mm}{\includegraphics[height=6mm]{tableaux/6_611223355446}}\
%\in Y_6(256431,4), \quad
, \quad
T =
\raisebox{-1.6mm}{\includegraphics[height=6mm]{tableaux/6_612354}}\ ,
\end{equation*}
with $U \in Y_6(246531,6)$, and $T$ having rightmost entry $4$.
Removing $6$ from $T$ and moving $4$ to the leftmost position,
we have the tableaux
\begin{equation*}
T' =
\raisebox{-1.6mm}{\includegraphics[height=6mm]{tableaux/5_41235}}\ , \quad
U' = 
\raisebox{-1.6mm}{\includegraphics[height=6mm]{tableaux/5_4112233554}}\ ,
\end{equation*}
with
$T' = L(U')$ and
$U' \in Y_5(F_{24531},4)$.

\begin{lem}\label{l:delta2}
For each connected zig-zag
%descending star 
network $F_w$ of order $n$, 
the map $\delta_2$ 
%(\ref{eq:delta2def}) 
is a bijection.  Furthermore, if $\delta_2(U) = U' \in Y_{n-1}(F_{\iota(w)},j)$ 
and $b$ is as in (\ref{eq:delta2})
%and the leftmost entry of $L(U')$ is $j$
then we have 
\begin{equation*}
\inv(U) = 
\begin{cases}
\inv(U') + n - b &\text{if $[b,n]$ maximal in $\preceq$},\\
\inv(U') + 2n - 2j - 1 &\text{otherwise}.
%&\text{if $[b,n]$ minimal in $\preceq$},\\
\end{cases}
\end{equation*}
\end{lem}
\begin{proof}
To invert $\delta_2$, let $U'$ be an element of 
%$Y(F_{\iota(w),1}) \cup \cdots \cup Y(F_{\iota(w)},n-1)$.
$Y(F_{\iota(w)},j)$ for some $1 \leq j \leq n-1$.
Create a Young tableau $T$
from $L(U')$ by inserting $n$ into the leftmost position
and if $[b,n]$ is not maximal in $\preceq$ 
by moving $j$ to the rightmost position.
%($\star$ Why is this a valid $F_w$-tableau?)
By Lemma~\ref{l:ndelins} (2) there is a unique $F_w$-tableau 
$U = (\pi_{i_1}, \dotsc, \pi_{i_n})$ satisfying $L(U) = T$.

To compare inversions in $U$ and $U'$, 
observe first that the paths
$\pi_{i_2}, \dotsc, \pi_{i_{n-1}}$ in $U$ have the same 
sources and sinks as the corresponding paths in 
\begin{equation*}
U' = 
\begin{cases}
(\pi'_{i_2}, \dotsc, \pi'_{i_n}) &\text{if $[b,n]$ is maximal in $\preceq$},\\
(\pi'_{i_n} = \pi'_j, \pi'_{i_2}, \dotsc, \pi'_{i_{n-1}}) &\text{otherwise}.
\end{cases}
\end{equation*}
Thus by Lemma~\ref{l:ndelins} (3), two such paths form an inversion
in $U$ if and only if the corresponding paths form an inversion in $U'$.
Consider therefore inversions in $U$ which involve the paths $\pi_{i_1} = \pi_n$
or $\pi_{i_n}$, and inversions in $U'$ which involve the path $\pi'_{i_n}$.

%suppose first that $[b,n]$ is maximal in $\preceq$
%and let $U' = (\pi'_{i_1}, \dotsc, \pi'_{i_n})$.
%Then paths 

If $[b,n]$ is maximal in $\preceq$, the path $\pi_{i_1} = \pi_n$
in $U$ terminates at sink $i_2 \in [b,n-1]$.
% with $b \leq i_2 \leq n-1$.  
Thus the path, which precedes all others in $U$, intersects only those
$n - b$ other paths in $U$ which terminate at sinks $[b,n] \ssm \{ i_2 \}$,
and thus also pass through the vertex of $F_w$
corresponding to the central vertex of $G_{[b,n]}$.
% forming $b - n$ inversions.
Now observe that $\pi_{i_n}$ terminates at sink $n$ of $F_w$, while
$\pi'_{i_n}$ terminates at sink $i_2$ of $F_{\iota(w)}$.  Since 
%$i_2 > b$,
$i_2 \geq b$,
%On ther other hand,
%we have $i_2 \geq b$ since $\pi_{i_1}$ terminates at sink $i_2$ of $F_w$.
%It follows 
%that 
the paths $\pi_{i_n}$ and $\pi'_{i_n}$ are identical
up to the vertex of $F_w$ ($F_{\iota(w)}$) which corresponds to the central
vertex of $G_{[b,n]}$.  Thus any path $\pi_{i_k}$ in $U$ intersects $\pi_{i_n}$
if and only if the corresponding path $\pi'_{i_k}$ in $U'$ intersects
$\pi'_{i_n}$.  It follows that $\inv(U) = \inv(U') + b - n$ in this case.
%Now suppose that 

If $[b,n]$ is not maximal in $\preceq$, then it is minimal.
%and let $U' = (\pi'_{i_n} = \pi'_j, \pi'_{i_2}, \dotsc, \pi'_{i_{n-1}})$.
%Again, paths $\pi_{i_2}, \dotsc, \pi_{i_{n-1}}$ in $U$ have the same 
%sources and sinks as the corresponding paths in $U'$.  Thus the 
%sets contribute equally to $\inv(U)$ and $\inv(U')$.
%Consider therefore inversions in $U$ which involve the paths $\pi_{i_1} = \pi_n$
%or $\pi_{i_n}$, and inversions in $U'$ which involve the path $\pi'_{i_n}$.
Since $\pi_{i_n}$ terminates at sink $n$,
% and since the interval $[b,n]$ is 
%minimal in $\preceq$, 
we have that $i_n \geq b$.  Thus $\pi_{i_n}$ intersects
and follows the $n - i_n + 1$ paths $\pi_{i_n + 1}, \dotsc, \pi_n$ which all 
intersect at least at the vertex of $F_w$ (or $F_{\iota(w)}$) corresponding 
to the central vertex of $G_{[b,n]}$. Now observe that the paths $\pi_n$
and $\pi'_{i_n}$ both terminate at sink $i_2$.  Thus, since 
%$i_n > b$,
$i_n \geq b$,
the paths are identical from the vertex of $F_w$ ($F_{\iota(w)}$) which 
corresponds to the central vertex of $G_{[b,n]}$ until sink $i_2$.  Thus
any path $\pi_k$ in $U$ intersects $\pi_n$ if and only if the corresponding
path $\pi'_k$ in $U'$ intersects $\pi'_{i_n}$.
It follows that $\inv(U) = \inv(U') + n - j + 1$ in this case.
%To see that the tableau $U$ satisfies
%$\inv(U) = \inv(U') + 2n - 2j - 1$, 
%let $(\pi_1, \dotsc, \pi_n)$ be the path family contained in $U$
%and let $(\pi'_1, \dotsc, \pi'_{n-1})$ be the path family contained in $U'$.
%Consider subsequences $(\pi_{\ell_2}, \pi_{\ell_1})$
%and $(\pi'_{\ell_2}, \pi'_{\ell_1})$ of paths in $U$ and $U'$, with 
%$\ell_1 < \ell_2 \leq n$.
%
%Suppose first that we have $\ell_2 \not\in \{ j, n \}$.
%If $\ell_1 \neq j$, then the source
%and sink indices of the paths $(\pi_{\ell_2}, \pi_{\ell_1})$
%are the same as those of the paths $(\pi'_{\ell_2}, \pi'_{\ell_1})$.
%Thus by Lemma~\ref{l:ndelins} (3),
%the pair $(\pi_{\ell_2}, \pi_{\ell_1})$ is an inversion in $U$ 
%if and only if 
%the pair $(\pi'_{\ell_2}, \pi'_{\ell_1})$ is an inversion in $U'$.
%If $\ell_1 = j$, then $(\pi'_{\ell_2}, \pi'_{\ell_1})$ is not an inversion in $%U'$,
%since $\pi'_{\ell_1}$ is the leftmost entry.
%On the other hand,
%$(\pi_{\ell_2}, \pi_{\ell_1})$ is an inversion in $U$ since 
%any two paths in $F_w$ whose source indices belong to $[j, n-1]$ intersect.  
%(Since $\pi_j$ terminates at sink $n$, we have 
%$j \geq c_1$, where $c_1$ is defined by the concatenation (\ref{eq:concat2})
%corresponding to $F_w$.)
%There are $n - j - 1$ such inversions in $U$.
%
%Now consider $\ell_2 \in \{ j, n \}$.
%Observe that $(\pi_j, \pi_{\ell_1})$ is not an inversion in $U$
%since $\pi_j$ is the rightmost entry,
%but $(\pi'_j, \pi_{\ell_1})$ may be an inversion in $U'$. 
%There is no path $\pi'_n$ in $U'$, but $(\pi_n, \pi_{\ell_1})$ may be an
%inversion in $U$.
%
%Suppose that $\ell_1 < j$ 
%and let $m$ be the common index of the sinks of $\pi_n$ in $F_w$
%and $\pi'_j$ in $F_{\iota(w)}$.
%We claim that inversions $(\pi_n, \pi_{\ell_1})$ in $U$
%correspond bijectively to 
%inversion $(\pi'_j, \pi'_{\ell_1})$ in $U'$.
%By Lemma~\ref{l:dsnintersect}, the sequence
%$(\pi_n, \pi_{\ell_1})$ is an inversion in $U$
%if and only if
%there is a path in $F_w$ from source $\ell_1$ to sink $m$.
%By Observation~\ref{o:sourcetosink}, this path exists if and only if
%$\ell_1 > m$, or $\ell_1$ and $m$ belong to a common interval in the
%concatenation (\ref{eq:concat2}) corresponding to $F_w$.
%Equivalently, by the definition of $\iota(w)$,
%$\ell_1 > m$ or $\ell_1$ and $m$ belong to a common interval in the
%concatenation (\ref{eq:iotadef}) corresponding to $F_{\iota(w)}$.
%By Observation~\ref{o:sourcetosink} again, this condition holds
%if and only if
%there is a path in $F_{\iota(w)}$ from source $\ell_1$ to sink $m$.
%By Lemma~\ref{l:dsnintersect} again, this is equivalent to
%$(\pi'_j, \pi'_{\ell_1})$ being an inversion in $U'$.
%
%Now suppose that $\ell_1 \geq j$.  Then 
%$(\pi'_j, \pi'_{\ell_1})$ is not an inversion in $U'$, but
%$(\pi_n, \pi_{\ell_1})$ is an inversion in $U$ since
%any two paths in $F_w$ whose source indices belong to $[j, n]$ intersect.  
%There are $n - j$ such inversions in $U$.
\end{proof}

%, and $\inv(U') = 3$.
%In the concatenation corresponding to $F_{256431}$, we have $c_1 = 3$,
%and in $T$, the number of entries in $[3,5]$ to the right of $6$ is $2$.

%($\star$ later?)
Note that in the example preceding Lemma~\ref{l:delta2},
%the unique interval containing $m = 6$ in 
the network $F_{246531}$ (\ref{eq:F246531}) of order $m = 6$ 
corresponds to the concatenation $G_{[3,6]} \circ G_{[2,4]} \circ G_{[1,2]}$,
and the interval $[3,6]$ is not maximal in the poset 
$[3,6] \prec [2,4] \prec [1,2]$.
The tableaux $U$, $U'$ satisfy
%we have $m = 6$, 
$j = 4$, $\inv(U) = 6$, $\inv(U') = 3$, and
%the tableaux $U$ and $U'$ satisfy
%\begin{equation*}
$\inv(U) = \inv(U') + 2m - 2j - 1$.
%\end{equation*}
%later

Now we return to the problem of factoring the inner sums in (\ref{eq:sumxu}).

\begin{prop}
Fix $w \in \mfs m$ \avoidingp{} 
%the pattern $312$ 
and an index $j \in [m]$.
%Fix $w \in \sn$ \avoidingp, 
%$\lambda  = (\lambda_1,\dotsc,\lambda_r) \vdash n$,
%$\lambda  = (\lambda_1,\dotsc,\lambda_r) \vdash m$,
%and a set partition $X = (X_1,\dotsc,X_r)$ of type $\lambda$.
%Then for $i = 1, \dotsc, r$ and $j \in [\lambda_i]$ we have
%Each sum $\sum_{U^{(i)}}q^{\inv{U^{(i)}}}$ on the right-hand side of 
%Equation~(\ref{eq:multisum}) is equal to 
Then we have
\begin{equation}\label{eq:qjm1}
%\sum_{U^{(i)}} \in Y_{\lambda_i}(F_w|_{X_i},j)}q^{\inv(U^{(i)})} = q^{j-1}O(F_w|_{X_i})
\sum_{U \in Y_m(F_w,j)}q^{\inv(U)} = q^{j-1}O(F_w).
\end{equation}
%i.e., 
%the sum is 
%%over cylindrical $F_w|_{X_i}$-tableaux $U^{(i)}$ 
%over cylindrical $F_v$-tableaux $U$ 
%%of shape $(\lambda_i)$ such that the tableau $L(U^{(i)})$ 
%of shape $(m)$ such that the tableau $L(U)$ 
%%begins with its $j$th least entry.
%begins with $j$.
%%of appears first.
%Equivalently, we have
%\begin{equation}\label{eq:qqanalog}
%%\sum_{U^{(i)}}q^{\inv(U^{(i)})} = [\lambda_i]_qO(F_w|_{X_i})
%\sum_{U}q^{\inv(U)} = [m]_qO(F_v)
%\end{equation}
%%for $i = 1, \dotsc, r$, 
%where the sum is 
%%over all cylindrical $F_w|_{X_i}$-tableaux $U^{(i)}$ 
%where the sum is over all cylindrical $F_v$-tableaux $U$ 
%%of shape $(\lambda_i)$.
%of shape $(m)$.
\end{prop}
\begin{proof}
We prove (\ref{eq:qjm1}) by induction on $m$.
% = \lambda_i$.  
%Observe that when $|X_i| = m$, the descending star network
%$F_w|_{X_i}$ is isomorphic to $F_v$ for some $v \in \mfs m$.
%a descending star network of order $m$
The only zig-zag 
%descending star 
network of order $m = 1$ is $F_e$, $e \in \mfs 1$.
Thus the set $Y_1(F_e, 1)$ 
%we have $v = e$, $j = 1$, and 
%the set $Y_1(F_w|_{X_i},1)$ 
%the set $Y_1(F_e,1)$ 
consists of one tableau of shape $(1)$ having no inversions.
%containing one path and therefore having no inversions.  
%whose source and sink indices are equal to $\{ X_i \}$.  
The left- and right-hand sides of (\ref{eq:qjm1}) are therefore 
$q^0 = 1$
%$1$
%, 
and
%the right-hand side is 
$q^0O(F_e) = [0]_q! = 1$, respectively.
%$, 
%$e \in \mfs 1$, 
%which is $[0]_q! = 1$.

Now assume (\ref{eq:qjm1}) to hold for zig-zag
%descending star 
networks corresponding to \pavoiding
%$312$-avoiding 
permutations in $\mfs {m-1}$
%blocks of $X$ having cardinality at most
%$m-1$, and consider a block $X_i$ of cardinality $m$ and an integer $j \in [m]$.
%permutations in $\mfs{m-1}$ 
%\avoidingp,
%and consider a permutation $w \in \mfs m$ whose planar network $F_w$ corresponds
%to the concatenation (\ref{eq:concat2}), and an integer $j \in [m]$.
and consider $F_w$ with $w \in \mfs m$ \avoidingp.
% $312$.
Let $F_w$ correspond to the concatenation (\ref{eq:concat2}) 
%$G_{[c_1,d_1]} \circ \cdots \circ G_{[c_t,d_t]}$
%(\ref{eq:concat2}) 
of star networks with $[b,m]$ the unique interval containing $m$,
let $u = \iota(w)$, and fix an integer $j \in [m]$.
If $F_w$ is disconnected, then
the set $Y_m(F_w,j)$ is empty and both sides of (\ref{eq:qjm1}) are $0$.
%the polynomial $O(F_w|_{X_i})$ is $0$,
%and the claim holds.
Assume therefore that $F_w$ is connected.
%and 
%let $F_u$ be the descending star network of order $m-1$ obtained by
%deleting the path $\pi_n$ from $F_v$. 
%Suppose first that $j \neq m$.  Then 
If $j < m$, then 
by Lemma {\ref{l:delta1}} 
and induction 
we have
\begin{equation}\label{eq:mminusc11}
%\begin{aligned}
\sum_{U \in Y_m(F_w,j)}q^{\inv(U)} 
= \sum_{k=0}^{m-b-1} q^k \sum_{U' \in Y_{m-1}(F_u,j)}q^{\inv(U')}\\
= [m-b]_q q^{j-1}O(F_u).
\end{equation}
Similarly, if $j = m$, then 
by Lemma {\ref{l:delta2}} 
and induction 
we have
\begin{equation}\label{eq:mminusc12}
\begin{aligned}
\sum_{U \in Y_m(F_w,m)}q^{\inv(U)} 
&= \begin{cases}
\displaystyle{\sum_{k=b}^{m-1}} q^{m-b} 
%\sum_{U' \in Y_{m-1}(F_u,k)}q^{\inv(U')
q^{k-1} O(F_u) 
%= [m-b]_q q^{m-1} O(F_u)
&\text{if $[b,m]$ maximal in $\preceq$},\\
\displaystyle{\sum_{k=b}^{m-1}} q^{2m-2k-1} 
q^{k-1} O(F_u) 
%= [m-b]_q q^{m-1} O(F_u)
%\sum_{U' \in Y_{m-1}(F_u,k)}q^{\inv(U')}
&\text{otherwise},
\end{cases}\\
%\Huge{\}}
&= [m-b]_q q^{m-1} O(F_u).
\end{aligned}
\end{equation}
%By induction, the inner sums are both $q^{k-1} O(F_u)$.
%\begin{equation*}
%%= \sum_{k=c_1}^{m-1} q^{2m-2k-1} q^{k-1} O(F_u)
%= \sum_{k=c_1}^{m-1} q^{2m-k-2} O(F_u).
%\end{equation*}
%Setting $\ell = m-1-k$ we see that this expression is equal to
%Both of these expressions are 
%which in both cases is equal to
%\begin{equation}\label{eq:mminusc12}
%\sum_{\ell=0}^{m-1-c_1} q^{m-1+\ell} O(F_u) = 
%$[m-b]_q q^{m-1} O(F_u)$.
%\end{equation}
Now 
%by
applying 
%Equation
% (\ref{eq:QF}), 
%(\ref{eq:iotadef})
(\ref{eq:OandO})  
to (\ref{eq:mminusc11}), (\ref{eq:mminusc12}),
%we have
%that 
%the left-hand side of (\ref{eq:qjm1}) equals 
we have for $j = 1, \dotsc, m$ that 
%\end{equation*}
%If $d_2 < m-1$ in (\ref{eq:concat2}), then this expression is equal to
\begin{equation*}
%\begin{aligned} 
\sum_{U \in Y_m(F_w,j)}q^{\inv(U)} = 
q^{j-1}[m-b]_q O(F_u) = q^{j-1} O(F_w).
%\\
%= \begin{cases}
%q^{j-1}
%[m-c_1]_q 
%\displaystyle{
%\frac{[m-1-c_1]_q! [d_2 - c_2]_q! \cdots [d_t - c_t]_q!}
%{[d_2 - c_1]_q! [d_3 - c_2]_q! \cdots [d_t - c_{t-1}]_q!}}
%&\text{if $d_2 < m-1$},\\
%\displaystyle{\frac{q^{j-1}[m-c_1]_q!}{[m-1-c_1]_q!}} 
%\displaystyle{
%\frac{[d_2 - c_2]_q! \cdots [d_t - c_t]_q!}
%{[d_3 - c_2]_q! \cdots [d_t - c_{t-1}]_q!}}
%&\text{if $d_2 = m-1$}.
%\end{cases}
%\end{aligned}
\end{equation*}
%for $j = 1, \dotsc, m$.
%By 
%(\ref{eq:OFdef}) 
%(\ref{eq:OandO}) 
%this is $q^{j-1}O(F_w)$.
%If $j = m$, then
%by Lemma {\ref{l:delta2}} we have
%, and let $U$ be a cylindrical $F_v$-tableau 
%of shape $(m)$ with the first entry of $L(U)$ being less than $m$.
%Let $k$ be the number of entries less than $m$ and appearing to the right
%of $m$ in $L(U)$.
%Let $F_u$ be the descending star network of order $m-1$ obtained by
%deleting the path $\pi_n$ from $F_v$, 
% to obtain a descending star network $F_u$,
%and let $U'$ be the cylindrical tableau of shape $(m-1)$ obtained from $U$ by
%deleting the cell containing $\pi_n$. 
%By 
%left-hand side of Equation (\ref{eq:qjm1} 
%is trivially true.
\end{proof}

%($\star$ Now explain why $X_i$ having $\lambda_i$ elements corresponds
%to $F_v$ with $v \in \mfs {\lambda_i}$.)
%($\star$ Is there an unanchored version of this result corresponding
%to the unanchored factorization in the earlier question?)

Applying (\ref{eq:CY}) 
%($\star$ that result wasn't exactly true)
to the previous result, we have the following.
\begin{cor}\label{c:SOF}
Let $w \in \mfs m$ \avoidp.
% the pattern $312$.  
Then we have
\begin{equation*}
\sum_{U \in \mathcal C([m],F_w)} q^{\inv(U)} = [m]_q O(F_w),
\qquad
\sum_{U \in \mathcal C_L([m],F_w)} q^{\inv(U)} = O(F_w).
%\sum_{U \in \mathcal C([n],F_w)} q^{\inv(U_r \circ \cdots \circ U_1)} = [n]_q O(F_w).
\end{equation*}
\end{cor}

%Generalizing this result is the following formula, where
%the sum is now over all tableaux in $\mathcal C(I, F_w)$.
%\begin{cor}\label{c:SOF2}
%Fix $w \in \sn$ avoiding the pattern $312$, 
%$\lambda  = (\lambda_1,\dotsc,\lambda_r) \vdash n$,
%and a set partition $I = (I_1,\dotsc,I_r) \vdash [n]$ of type $\lambda$.
%Then we have
%\begin{equation*}
%\sum_{U \in \mathcal C(I,F_w)} q^{\inv(U_r \circ \cdots \circ U_1)} = 
%[\lambda_1]_q \cdots [\lambda_r]_q
%\sum_{U \in \mathcal C_L(I,F_w)} q^{\inv(U_r \circ \cdots \circ U_1)}.
%\end{equation*}
%\end{cor}
%\begin{proof}
%Clearly there is a bijection between $\mathcal C(I,F_w)$ and 
%$[\lambda_1] \times \cdots \times [\lambda_r] \times \mathcal C_L(I_,F_w)$
%given by $U \mapsto (a_1,\dotsc,a_r, U')$, where
%$a_k$ is the position in row $k$ of the path having the least index
%among all paths in row $k$, and $U'$ is obtained from $U$ by
%cyclically shifting each row so that the path of least index appears first.
%\end{proof}

%\begin{prop}
%The expression in Equation (\ref{eq:sumxu}) is equal to
%\begin{equation*}
%\sum_X q^{\inv(V(X))} O(F_w|_{X_1}) \cdots O(F_w|_{X_r}),
%\end{equation*}
%where the sum is over all ordered set partitions of $[n]$.
%\end{prop}
%\begin{proof}
%Omitted.
%\end{proof}

Now we have the following $q$-analogs of 
Theorem~\ref{t:sncfinterp} (iv-a).
\begin{thm}\label{t:qpsi}
Let $w \in \sn$ \avoidp.
For $\lambda = (\lambda_1, \dotsc, \lambda_r) \vdash n$, we have
\begin{equation}\label{eq:qpsi1}
\psi_q^\lambda(\qew C'_w(q)) 
= \sum_U q^{\inv(U_1 \circ \cdots \circ U_r)},
%= \sum_U q^{\inv(U_r \circ \cdots \circ U_1)},
\end{equation}
where the sum is over all cylindrical $F_w$-tableaux of shape $\lambda$, and
\begin{equation}\label{eq:qpsi2}
\psi_q^\lambda(\qew C'_w(q)) 
= [\lambda_1]_q \cdots [\lambda_r]_q \sum_U q^{\inv(U_1 \circ \cdots \circ U_r)},
%= [\lambda_1]_q \cdots [\lambda_r]_q \sum_U q^{\inv(U_r \circ\cdots\circ U_1)}
\end{equation}
where the sum is over all left-anchored cylindrical $F_w$-tableaux 
of shape $\lambda$.
%and
%\begin{equation}\label{eq:qpsi3}
%\begin{aligned}
%\psi_q^\lambda(\qew C'_w(q)) 
%&= [\lambda_1]_q \cdots [\lambda_r]_q
%\sumsb{I \vdash [n]\\ \type(I) = \lambda}
%q^{\inv(V(F_w,I)_1 \circ \cdots \circ V(F_w,I)_r)} O(F_w|_{I_1}) \cdots O(F_w|_%{I_r})\\
%&= [\lambda_1]_q \cdots [\lambda_r]_q
%\sumsb{I \vdash [n]\\ \type(I) = \lambda}
%q^{\inv(V(F_w,I)_r \circ \cdots \circ V(F_w,I)_1)} O(F_w|_{I_1}) \cdots O(F_w|_%{I_r}).
%\end{aligned}
%\end{equation}
\end{thm}
\begin{proof}
Rewrite the 
%right-hand sides
sums 
%in (\ref{eq:qpsi1}), (\ref{eq:qpsi2})
above 
as in 
%(\ref{eq:}), (\ref{eq:})
(\ref{eq:sumxu}) and factor the resulting inner sums
%in both cases 
as in
Proposition~\ref{p:multisum}.
By Corollary~\ref{c:SOF}, 
%we obtain
the right-hand sides of 
(\ref{eq:qpsi1}), (\ref{eq:qpsi2}) are both equal to
%are equal to
%we now have
\begin{equation*}
%\begin{gathered}
%\psi_q^\lambda(\qew C'_w(q)) = 
%\sumsb{I \vdash [n]\\ \type(I) = \lambda} 
%q^{\inv(V(F_w,I)_1 \circ \cdots \circ V(F_w,I)_r)}
%[\lambda_1]_q O(F_w|_{I_1}) \cdots [\lambda_r]_q O(F_w|_{I_r}),\\
[\lambda_1]_q \cdots [\lambda_r]_q
\sumsb{I \vdash [n]\\ \type(I) = \lambda} 
q^{\inv(V(F_w,I)_1 \circ \cdots \circ V(F_w,I)_r)}
O(F_w|_{I_1}) \cdots O(F_w|_{I_r}).
%\end{gathered}
\end{equation*}
%respectively.
By Theorem~\ref{t:qpsiO}, this is $\psi_q^\lambda(\qew C'_w(q))$.
\end{proof}


%($\star$ Do an example.)
For example, consider the descending star network 
$F_{3421}$ in (\ref{eq:F3421}) 
%and the first sum in 
and the sum in (\ref{eq:qpsi1}).
It is easy to verify that there are eighteen 
cylindrical $F_{3421}$-tableaux of shape $31$.
Four of these are
\begin{equation}\label{eq:cyltabs31}
\raisebox{-6mm}{\includegraphics[height=12mm]{tableaux/31_11433224}}\ ,\quad
\raisebox{-6mm}{\includegraphics[height=12mm]{tableaux/31_22133441}}\ ,\quad
\raisebox{-6mm}{\includegraphics[height=12mm]{tableaux/31_33122441}}\ ,\quad
\raisebox{-6mm}{\includegraphics[height=12mm]{tableaux/31_44233112}}\ ,
\end{equation}
where $i,j$ represents the unique path from source $i$ to sink $j$.
These tableaux $U$ of shape $31$
%and if we construct from each tableau $U$ 
%yield tableaux $U_2 \circ U_1$ of shape $4$,
yield tableaux $U_1 \circ U_2$ of shape $4$,
\begin{equation}\label{eq:cyltabs4}
%\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_11433224}}\ ,\quad
%\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_22133441}}\ ,\quad
%\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_33122441}}\ ,\quad
%\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_44233112}}\ ,
\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_43322411}}\ ,\quad
\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_13344122}}\ ,\quad
\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_12244133}}\ ,\quad
\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_23311244}}\ ,
\end{equation}
%which have $3$, $1$, $2$, and $4$ inversions, respectively.
which have $5$, $2$, $1$, and $2$ inversions, respectively.
Together, 
they
%the tableaux 
%contribute $q + q^2 + q^3 + q^4$ 
contribute $q + 2q^2 + q^5$ 
to $\psi_q^{31}(q_{e,3421} C'_{3421}(q)) = 1 + 3q + 5q^2 + 5q^3 + 3q^4 + q^5$.
%Similarly,
%The reader can verify that the same four tableaux of shape $31$
%yield tableaux $U_2 \circ U_1$ of shape $4$
%which have $3$, $1$, $2$, and $4$ inversions, respectively.
Now consider the sum in (\ref{eq:qpsi2}).
It is easy to verify that there are six left-anchored 
cylindrical $F_{3421}$-tableaux of shape $31$: the second and third tableaux
in (\ref{eq:cyltabs31}) and the four tableaux
%Four of these are
\begin{equation*}
%\label{eq:lacyltabs31}
\raisebox{-6mm}{\includegraphics[height=12mm]{tableaux/31_11233442}}\ ,\quad
\raisebox{-6mm}{\includegraphics[height=12mm]{tableaux/31_11244332}}\ ,\quad
\raisebox{-6mm}{\includegraphics[height=12mm]{tableaux/31_44122331}}\ ,\quad
\raisebox{-6mm}{\includegraphics[height=12mm]{tableaux/31_44133221}}\ .
\end{equation*}
These tableaux $U$ of shape $31$
%yield six tableaux $U_2 \circ U_1$ of shape $4$: 
yield six tableaux $U_1 \circ U_2$ of shape $4$: 
the second and third tableaux in (\ref{eq:cyltabs4}) and the four tableaux
\begin{equation*}
%\label{eq:lacyltabs4}
%\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_11233442}}\ ,\quad
%\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_11244332}}\ ,\quad
%\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_44122331}}\ ,\quad
%\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_44133221}}\ ,
\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_23344211}}\ ,\quad
\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_24433211}}\ ,\quad
\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_12233144}}\ ,\quad
\raisebox{-3mm}{\includegraphics[height=6mm]{tableaux/4_13322144}}\ ,
\end{equation*}
%which have $1$, $2$, $0$, $1$, $2$, and $3$ inversions, respectively.
which have $2$, $1$, $2$, $3$, $0$, and $1$ inversions, respectively.
Together, 
%they
the six tableaux 
contribute $1 + 2q + 2q^2 + q^3$
%) = 1 + 3q + 5q^2 + 5q^3 + 3q^4 + q^5$.
to 
%$[3]_q [1]_q (1 + 2q + 2q^2 + q^3) = 1 + 3q + 5q^2 + 5q^3 + 3q^4 + q^5$.
%to 
%$\psi_q^{31}(q_{e,3421} C'_{3421}(q)) = 
$[3]_q [1]_q (1 + 2q + 2q^2 + q^3) 
%= 1 + 3q + 5q^2 + 5q^3 + 3q^4 + q^5
= \psi_q^{31}(q_{e,3421} C'_{3421}(q))$. 
%The reader can verify that the same six tableaux of shape $31$
%yield six different tableaux $U_1 \circ U_2$ of shape $4$
%which have $1$, $2$, $0$, $1$, $2$, and $3$ inversions, respectively.

%($\star$ Introduce this.)
As a consequence of Theorem~\ref{t:qpsi}, we now have the following analog
of Corollary~\ref{c:edeck}.
Say that a partition $\lambda$ of $n$ is a {\em refinement}
of another partition (or composition) $\mu$ of $n$ if 
$\lambda$ can be obtained from $\mu$ by replacing each part $\mu_i$ 
by an integer partition of $\mu_i$
and sorting the results into weakly decreasing order.
%($\star$ Is this a theorem now?)
%($\star$ Make $\mu$ a composition above and below.)

\begin{cor}\label{c:prefine}
Let $w \in \sn$ \avoidp and let the component sizes of $F_w$ 
%(in weakly decreasing order) 
be $\mu = (\mu_1, \dotsc, \mu_s)$. 
%Then for each partition $\lambda \vdash n$ not refining $\mu$ we have
Then we have
\begin{equation*}
\psi_q^\lambda (\qew C'_w(q)) = 0.
\end{equation*}
%unless $\lambda$ is a refinement of $\mu$.
if and only if $\lambda$ is not a refinement of $\mu$.
\end{cor}
\begin{proof}
It is clear that if $\lambda$ is not a refinement of $\mu$,
then there is no cylindrical $F_w$-tableau of shape $\lambda$.
Therefore by Theorem~\ref{t:qpsi} we have that $\psi_q^\lambda(\qew C'_w(q)) = 0$.
Suppose on the other hand that $\lambda = (\lambda_1, \dotsc, \lambda_r)$ 
is a refinement of $\mu$, and let $J_1, \dotsc, J_s$ be the
subintervals of $[n]$ corresponding to the connected components of $F_w$.
Then there exists an ordered set partition
$I = (I_1, \dotsc, I_r) \vdash [n]$,
%$J = (J_1, \dotsc, J_s)$
%of $[n]$ 
whose type is a rearrangement of $\lambda$,
%, consisting entirely of subintervals,
%with the properties
%\begin{enumerate}
%\item Each block of $I$ 
%is a subinterval of $[n]$ having 
%has cardinality
%$\lambda_i$ for some $i$.
%\item Each 
such that each block of $J$ is a union of several consecutive blocks of $I$.
%\item Each block of $J$ 
%is a subinterval of $[n]$ 
%corresponds to a connected component $F_w|_{J_i}$ of $F_w$. 
%\end{enumerate}
Let $\pi = (\pi_1, \dotsc, \pi_n)$ be the unique path family of type $e$
covering $F_w$.
It is clear now that we can construct at least one record-free, row-semistrict
$F_w$ tableau of type $e$ and shape $\lambda$, by creating a row
containing the paths $\pi_a, \dotsc, \pi_b$ (in order)
for each block $[a,b]$ of $I$.
By Theorem~\ref{t:swqpsi}, we therefore have
$\psi_q^\lambda (\qew C'_w(q)) \neq 0$.
%consisting of the paths 
\end{proof}

It would be interesting to extend Theorem~\ref{t:qpsi}
to include a $q$-analog of Theorem~\ref{t:sncfinterp} (iv-b).
In particular the fourth identity in Equation~(\ref{eq:moncfdef})
suggests that an answer to Problem~\ref{p:monevalinterp} and its $q$-analog
are related to a set partition of tableaux counted by $\psi^\lambda$.
It is not clear whether such a partition is more easily expressed
in terms of 
record-free, row-semistrict $F_w$-tableaux of type $e$,
right-anchored, row-semistrict $F_w$-tableaux of type $e$,
cylindrical $F_w$-tableaux, 
left-anchored cylindrical $F_w$-tableaux, or
cyclically row-semistrict $F_w$-tableaux of type $e$.

\begin{prob}
Find a statistic $\STAT$ on $F$-tableaux such that we have
\begin{equation*}
\psi_q^\lambda(\qew C'_w(q)) = \sum_U q^{\STAT(U)},
\end{equation*}
where the sum is over all cyclically row-semistrict $F_w$-tableaux 
of type $e$ and shape $\lambda$.
\end{prob}

%($\star$) It appears that we can change the order of rows in the theorems.
It would also be interesting to show that all 
%of the 
stated
%our 
interpretations
of $\psi_q^\lambda(\qew C'_w(q))$ remain valid if we reverse the order of 
concatenating rows of tableaux: this may turn out to be the easiest way
to link the $\hnq$-traces $\psi_q^\lambda$ and $\phi_q^\mu$.

\begin{prob}
Show that Theorems~\ref{t:swqpsi}, \ref{t:qpsiO}, \ref{t:qpsi}
remain valid if one replaces the numbers
$\inv(U_1 \circ \cdots \circ U_r)$,
$\inv(U_1^R \circ \cdots \circ U_r^R)$,
$\inv(V(F_w,I)_1 \circ \cdots \circ V(F_w,I)_r)$,
%are replaced 
with
$\inv(U_r \circ \cdots \circ U_1)$, 
$\inv(U_r^R \circ \cdots \circ U_1^R)$,
$\inv(V(F_w,I)_r \circ \cdots \circ V(F_w,I)_1)$,
respectively.
%$U_1 \circ \cdots \circ U_r$,
%$U_1^R \circ \cdots \circ U_r^R$,
%$V(F_w,I)_1 \circ \cdots \circ V(F_w,I)_r$,
%are replaced with
%$U_r \circ \cdots \circ U_1$, 
%$U_r^R \circ \cdots \circ U_1^R$,
%$V(F_w,I)_r \circ \cdots \circ V(F_w,I)_1$,
%respectively.
\end{prob}

\section{Results concerning $\phi_q^\lambda(\qew C'_w(q))$}\label{s:hnqinterpm}

%However, there is no conjectured combinatorial interpretation 
%for $\chi_q^\lambda(\qew C'_w(q))$ unless $w$ avoids $312$.
%Haiman~\cite{HaimanHecke} 

%As mentioned in Section~\ref{s:intro}, there is no monomial-trace analog
Recall that the component statements of Theorem~\ref{t:sncfinterp} pertaining
to monomial traces are weaker 
%statements 
than those pertaining to other traces.
To state a $q$-analog of Theorem~\ref{t:sncfinterp} (v-a), we will use
several partial orders, including majorization and refinement of integer
partitions.  We will use the symbol $\unlhd$ to denote majorization and 
$\rleq$ to denote refinement, as defined before Corollary~\ref{c:prefine}.
%order on partitions.
We begin by stating an analog of Corollaries \ref{c:edeck}, \ref{c:sdeck}.

%(i)--(iv-d), but 
%Haiman~\cite{HaimanHecke} has conjectured 
%that we have $\phi_q^\lambda(\qew C'_w(q)) \in \mathbb N[q]$ 
%for all $\lambda \vdash n$ and all $w \in \sn$.
%There is no general conjectured combinatorial interpretation 
%for $\phi_q^\lambda(\qew C'_w(q))$, even in the case that $w$ avoids $312$.

\begin{prop}\label{p:mnodec}
Fix $\lambda = (\lambda_1, \dotsc, \lambda_r) \vdash n$
and $w \in \sn$ \avoidingp.  If $w_1 \cdots w_n$ has 
a decreasing subsequence of length greater than $\lambda_1$, then 
we have $\phi_q^\lambda(\qew C'_w(q)) = 0$.
\end{prop}
\begin{proof}
Let $w \in \sn$ have a decreasing subsequence of 
length greater than $\lambda_1$ and recall that there exist integers 
$\{a_{\lambda,\mu} \,|\, \lambda,\mu \vdash n\}$ such that
\begin{equation*}
\phi_q^\lambda = \sum_{\mu \unrhd \lambda^\tr} a_{\lambda,\mu} \epsilon_q^\mu.
%\phi_q^\lambda = \sum_{\mu \geq \lambda^\tr} a_{\lambda,\mu} \epsilon_q^\mu,
\end{equation*}
%where $\unlhd$ is the dominance order on partitions.
%where $\geq$ is the majorization order on partitions.
%(See \cite[p.\,288]{StanEC2}.)
%, e.g., 
%Section~\ref{s:intro} and 
%\cite{RemTrans}.)
%references there.)
If $\phi_q^\lambda(\qew C'_w(q)) \neq 0$ then some partition $\mu$
in the above sum satisfies $\epsilon_q^\mu(\qew C'_w(q)) \neq 0$.
But the number of parts of $\mu$ is necessarily less than or equal to 
$\lambda_1$.  This contradicts
%$\star$ Use expansion of $\epsilon_q^\lambda$ in terms of $\phi_q^\lambda$ and
Corollary~\ref{c:edeck}.
\end{proof}
%($\star$ State a stronger conjecture, using Greene-style subsequences,
%RS shapes of permutations, and/or material in the bitableaux paper?  
%Use the same strategy to strengthen Corollaries \ref{c:edeck}, \ref{c:sdeck}?
%Admit that those results are special cases of more general results which
%can be proven algebraically using KL representations?)

%($\star$ State a conjecture for the rectangular case.)

%($\star$ Define component sizes and refinement order of partitions.)

%($\star$ What $p$-version of the above result can we conjecture?)

%$\star$ State a conjecture for the case that $\lambda_1 \leq 2$.

%($\star$ Introduce this.)

Similarly, we have a partial analog of Corollary~\ref{c:prefine}.
%(This also follows from
%Alternatively, Proposition~\ref{p:mdeck} follows from 
%Haiman's 
%more general 
%formula \cite[Prop.\,4.1\,(2)]{HaimanHecke}.)
\begin{prop}\label{p:mdeck}
Let $w \in \sn$ \avoidp{} and let the component sizes of $F_w$ 
(in weakly decreasing order) be
$\mu = (\mu_1, \dotsc, \mu_r)$. 
Then for each partition $\lambda \vdash n$ not refining $\mu$ we have
\begin{equation*}
\phi_q^\lambda (\qew C'_w(q)) = 0.
\end{equation*}
%unless $\lambda$ is a refinement of $\mu$.
\end{prop}
\begin{proof}
%Let $\rleq$ be the refinement order on integer partitions of $n$
%and 
Observe that we may rewrite the last equation in (\ref{eq:moncfdef}) as
\begin{equation*}
\psi_q^\lambda = \sum_{\nu \geq_R \lambda} L_{\lambda,\nu} \phi_q^\nu,
\end{equation*}
since no row-constant Young tableau of shape $\lambda$ 
has content $\nu$ unless $\lambda$ refines $\nu$.  
Inverting the matrix $(L_{\lambda,\nu})_{\lambda,\nu \vdash n}$
and evaluating traces at $\qew C'_w(q)$ we have
\begin{equation}\label{eq:moncfinv}
\phi_q^\lambda(\qew C'_w(q)) = 
\sum_{\nu \geq_R \lambda} L_{\lambda,\nu}^{-1} \psi_q^\lambda(\qew C'_w(q)).
\end{equation}
Now suppose that we have $\lambda \not \rleq \mu$.
Then each partition $\nu$ in (\ref{eq:moncfinv}) 
satisfies $\nu \not \leq_R \mu$.
By Corollary~\ref{c:prefine}, each term on the 
right-hand side of (\ref{eq:moncfinv}) is zero.
\end{proof}

%Alternatively, 
We remark that Propositions~\ref{p:mnodec}, \ref{p:mdeck} are not new:
they follow from 
\cite[Prop.\,4.1]{HaimanHecke}.
%, respectively.
%\cite[Prop.\,4.1\,(3),\,(2)]{HaimanHecke}, respectively.
For more facts about these evaluations,
%$\phi_q^\lambda(\qew C'_w(q))$ 
see
%can be found in 
\cite[Sec\,4]{HaimanHecke}.

%This result follows from \cite[Prop.\,4.1\,(2)]{HaimanHecke}).
Now we complete the proof of
Theorem~\ref{t:sncfinterp} (v-a)
and provide a $q$-analog of this result.
%It will be convenient to let 
Let $\mathcal T_C(F_w,\mu)$ denote the set of column-strict $F_w$-tableaux of
shape $\mu$ and type $e$.
\begin{thm}\label{t:qphipartial}
Let $w \in \sn$ \avoidp.  For $\lambda_1 \leq 2$ we have
\begin{equation}\label{eq:qphipartial}
\phi_q^\lambda(\qew C'_w(q)) = 
\begin{cases}
%\displaystyle{\sum_U} q^{\inv(U)} 
\displaystyle{\sum_{U \in \mathcal T_C(F_w,\lambda)}} q^{\inv(U)} 
&\text{if for all $\mu \triangleleft \lambda$ we have
%majorized by $\lambda$ we have
$\mathcal T_C(F_w,\mu) = \emptyset$},\\  
0 &\text{otherwise}.
\end{cases}
\end{equation}
%with 
%Otherwise we have $\phi^\lambda(C'_w(1)) = 0$.
%where the sum is over all column-strict $F_w$-tableaux of shape $\lambda$
%and type $e$.
\end{thm}
\begin{proof}
%Clearly 
The claim is true for $\lambda = 1^n$, since 
%in this case
$\phi_q^{1^n} = \epsilon_q^{(n)}$
and the claimed formula coincides with that in Theorem~\ref{t:qepsilon}.
Suppose 
%therefore 
that the claim holds for 
$\lambda = 21^{n-2}, \dotsc, 2^{k-1}1^{n-2k+2}$
%$\lambda = 2^i 1^{n-2i}$, $i=0,\dotsc,k-1$ 
%and consider the case $\lambda = 2^k 1^{n-2k}$ 
and consider the case $\lambda = 2^k 1^{n-2k}$ 
%($i = k \leq \lfloor \tfrac n2 \rfloor$).
($k \leq \lfloor \tfrac n2 \rfloor$).
Then we have
\begin{equation}\label{eq:em12}
\epsilon_q^{(n-k,k)}(\qew C'_w(q)) 
= \sum_{i = 0}^{k} M_{2^k 1^{n-2k}, 2^i 1^{n-2i}} \phi_q^{2^i 1^{n-2i}} (\qew C'_w(q)),
\end{equation}
where $M_{\lambda,\mu}$ is the number of column-strict Young tableaux
of shape $\lambda$ and content $\mu$.
%More precisely, 
It is easy to see that
$M_{2^k 1^{n-2k}, 2^i 1^{n-2i}}$ is equal to 
$\tbinom{n-2i}{k-i}$.
By Theorem~\ref{t:qepsilon}, the left-hand side of (\ref{eq:em12}) is
the sum of $q^{\inv(U)}$ over $U \in \mathcal T_C(F_w,2^k 1^{n-2k})$.
%all column-strict $F_w$-tableaux of shape
%$2^k 1^{n-2k}$.
%($\star$ what?).
%($\star$ Continue next time.)

If $w$ has a decreasing subsequence of length three,
then by Proposition~\ref{p:mnodec}, 
the left-hand side of (\ref{eq:qphipartial}) is $0$.
By 
%the argument in 
the proof of Corollary~\ref{c:edeck},
we have $\mathcal T_C(F_w, \mu) = \emptyset$ for all 
$\mu \trianglelefteq \lambda$, and 
the right-hand side of (\ref{eq:qphipartial}) is $0$ as well.
%then $\mathcal T_C(F_w, \lambda)$ is empty
%and thus $\phi_q^\lambda(\qew C'_w(q)) = 0$ as claimed.

Assume therefore that $w$ avoids the pattern $321$.
%has no decreasing subsequence of length three.
By Theorem~\ref{t:kstar}, every connected component of $F_w$ induces
%corresponds to 
a subposet of $P(F_w)$ which is isomorphic to $P(H_k)$ where 
%$H_k$ is
%the planar network
\begin{equation*}
H_k = \raisebox{-12mm}{
%\raisebox{-8mm}{
\includegraphics[width=40mm]{xfigures/hk}}.
%\includegraphics[width=40mm]{2doublechains}}
\end{equation*}
Let $b = b(w)$ be the number of odd components of $F_w$.  
Then it is possible to construct an $F_w$-tableau which has
$b$ more paths in column $1$ than it has in column $2$, 
but it is not possible to construct an $F_w$-tableau 
for which this difference is greater than $b$.
%with more than this many ($\tfrac{n+b}2$) paths in column $1$.
That is, for $j = \frac{n-b}2$
%for some $i \leq \lfloor \frac n2 \rfloor$.
%by induction
we have $\mathcal T_C(F_w, 2^j 1^{n-2j}) \neq \emptyset$
% is nonempty, 
while
\begin{equation}\label{eq:emptysets}
\mathcal T_C(F_w, 1^{n}) = 
\mathcal T_C(F_w, 2 1^{n-2}) = \cdots =
\mathcal T_C(F_w, 2^{j-1} 1^{n-2j+2}) = \emptyset.
\end{equation} 
%is empty 
%for $i = 0,\dotsc, j-1$.
%\begin{equation}
%\phi_q^{2^i 1^{n-2i}} (\qew C'_w(q))
Fix $U \in \mathcal T_C(F_w, 2^j 1^{n-2j})$,
%$U$ be one such tableau, 
and let the interval $[p_1, p_{2m+1}]$
of sources and sinks define an odd component of $F_w$.
Then paths indexed by $p_1, p_3, \dotsc, p_{2m+1}$ belong
to the first column of $U$ while those indexed by $p_2,\dotsc,p_{2m}$
%all other paths in the interval
belong to the second column.  
Note that swapping the columns of the two sets of paths creates a 
valid column-strict $F_w$-tableau of shape $2^{j+1}1^{n-2j-2}$ 
which has the same number of inversions as $U$.
%sources and sinks 
Thus for each such tableau $U$, we may create a column-strict 
$F_w$-tableau $U'$ of shape $2^k 1^{n-2k}$ 
by choosing $k - j$ of the odd components and 
swapping the columns of the even and odd indexed paths within these components.
There are $\tbinom{n-2j}{k-j}$ ways to do this.
Conversely, every tableau $U' \in \mathcal T_C(F_w, 2^k 1^{n-2k})$ 
arises in this way.  
Thus we have
\begin{equation}\label{eq:emconnect}
\epsilon_q^{(n-k,k)}(\qew C'_w(q)) = 
\tbinom{n-2j}{k-j}
%\phi_q^{2^j 1^{n-2j}}(\qew C'_w(q)).
\sum_{U \in \mathcal T_C(F_w,2^j1^{n-2j})} q^{\inv(U)}.
\end{equation}
%$
%Thus we have
%\begin{equation*}

If $j < k$, then we have by induction and (\ref{eq:emptysets})
that
%, we have that 
$\phi_q^{2^{i} 1^{n-2i}}(\qew C'_w(q)) = 0$ for $i < k$, $i \neq j$.
%$i=0,\dotsc,j-1,j+1,\dotsc,k-1$. 
%$\phi_q^{2^{j+1} 1^{n-2j-2}}(\qew C'_w(q)) 
%= \cdots = 
%\phi_q^{2^{k-1} 1^{n-2k+2}}(\qew C'_w(q)) = 0$.
Now 
%the equalities (\ref{eq:em12}) -- (\ref{eq:emconnect}) imply that
(\ref{eq:emconnect}) implies that
$\phi_q^{2^k 1^{n-2k}}(\qew C'_w(q)) = 0$, and the claim is true.
% since $2^j1^{n-2j} \trianglelefteq 2^k1^{n-2k}$.
If $j = k$, then we have by (\ref{eq:em12}) -- (\ref{eq:emptysets}) 
%and induction
%? or what reason?)
that
%\begin{equation*}
$\epsilon_q^{(n-k,k)}(\qew C'_w(q)) = \phi_q^{2^k 1^{n-2k}}(\qew C'_w(q))$
%\end{equation*}
%and the claim is true.
and by (\ref{eq:emconnect}) the claim again is true.
%Since the restriction of $\trianglelefteq$ to partitions 
%$1^n, 21^{n-2}, \dotsc, 2^k1^{n-2k}$ is a total order,
%we also have by induction that $\phi_q^{2^i 1^{n-2i}}(\qew C'_w(q)) = 0$ 
%for $i = 0,\dotsc, j-1, j+1, \dotsc, k-1$.
\end{proof}

We remark that the obvious $q$-analogs of Theorem~\ref{t:sncfinterp} (v-b) are
false.  
%($\star$ Fill in the details.)
%Consider the partition $22$
Consider the permutation $w = 3142$
and the evaluation $\phi_q^{22}(\qew C'_w(q)) = q + q^2$.
$F_w$ is the penultimate zig-zag network 
in (\ref{eq:xfigureszz}), and 
%and the partition $22$.
%We have 
%, but the tableaux
there are two column-strict cylindrical $F_w$-tableaux of shape $22$,
\begin{equation*}\label{eq:cscyltabs22}
\raisebox{-6mm}{\includegraphics[height=12mm]{tableaux/22_43342112}}\ ,\quad
\raisebox{-6mm}{\includegraphics[height=12mm]{tableaux/22_34431221}}\ .
\end{equation*}
%partition $\lambda = (2,2)$.
Unfortunately, as $U$ varies over these tableaux we have
%summing $q^{\inv(U)}$, $q^{\inv(U_1 \circ U_2)}$, $q^{\inv(U_2 \circ U_1)}$ 
%over these tableaux gives
\begin{equation*}
\sum_U q^{\inv(U)} = 1 + q^3,
\qquad
\sum_U q^{\inv(U_1 \circ U_2)} = 1 + q^2,
\qquad
\sum_U q^{\inv(U_2 \circ U_1)} = q + q^3.
\end{equation*}
%two column-strict
%cylindrical $F_w$-tableaux of shape 22 
%satisfy
%the zig-zag network
%where the sum is over all column-strict, cylindrical $F_w$-tableaux
%of shape $22$.
Perhaps a correct $q$-analog of Theorem~\ref{t:sncfinterp} (v-b)
would help with the formulation of an interpretation of
$\phi_q^\lambda(\qew C'_w(q))$ when $w$ \avoidsp.
%We refer the reader to \cite[Sec\,4]{HaimanHecke} 
%for 
%($\star$ Try stating an analog of Theorem~\ref{t:sncfinterp} (v-a) for
%partitions with two or fewer rows, using refinement rather than majorization.)
%$\star$ Try combining $\epsilon^\lambda$ result 
%with Shareshian-Wachs conjectures for $\psi^\lambda$.
%Any extension of Theorem~\ref{t:} would be interesting.
%\begin{prob}\label{p:monevalinterpq}
%For (special cases of) $w \in \sn$ and $\lambda \vdash n$, 
%find a combinatorial proof that $\phi^\lambda(\qew C'_w(q))$ belongs
%to $\mathbb N[q]$.
%\end{prob}
Given Theorems \ref{t:qeta}, \ref{t:qepsilon}, \ref{t:qchi}, \ref{t:swqpsi}, 
\ref{t:qpsiO}, and \ref{t:qpsi},
it seems reasonable to hope that $F_w$-tableaux can 
play an important role in such an interpretation.
%aid in the interpretation
%of $\phi_q(\qew C'_w(q))$ for $w$ \avoidingp.
\begin{prob}\label{p:qphi}
Find a property $X$ of $F_w$-tableaux and a statistic $\STAT$ 
such that for $\lambda \vdash n$
and $w$ \avoidingp\ we have
\begin{equation*}
\phi_q^\lambda(\qew C'_w(q)) = \sum_U q^{\STAT(U)},
\end{equation*}
where the sum is over all $F_w$-tableaux $U$ of shape $\lambda$
having property $X$.
\end{prob}

\subsection*{Acknowledgements}

The authors are grateful to Kaitlyn Peterson, Daniel Studenmund, and Michelle
Wachs for helpful conversations, to Lehigh University, 
the University of Miami, and Universidad de los Andes for financial support
and hospitality.  The authors are also grateful to an anonymous referee
for helpful comments.

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\end{document}

