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\title{\bf Kirillov's Unimodality Conjecture for the Rectangular Narayana Polynomials}

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\author{Herman Z. Q. Chen\\
\small School of Science\\[-0.8ex] 
\small Tianjin Chengjian University, Tianjin 300384, P. R. China \\[-0.8ex] 
\small\tt zqchern@163.com 
\and
Arthur L. B. Yang\\
\small Center for Combinatorics, LPMC  \\[-0.8ex] 
\small Nankai University, Tianjin 300071, P. R. China \\[-0.8ex] 
\small\tt yang@nankai.edu.cn
\and
Philip B. Zhang\thanks{This work was supported by the National Science Foundation of China (Nos. 11231004, 11522110, 11626172, 11701424). 
%	The first author  was partially supported by the PHD Program  of Tianjin Chengjian University (160501436). The third author  was partially supported by the PHD Program  of Tianjin Normal University (52XB1616).
}\\
\small College of Mathematical Science \\[-0.8ex] 
\small Tianjin Normal University, Tianjin  300387, P. R. China \\[-0.8ex] 
\small\tt zhangbiaonk@163.com
}

% \date{\dateline{submission date}{acceptance date}\\
% \small Mathematics Subject Classifications: comma separated list of
% MSC codes available from http://www.ams.org/mathscinet/freeTools.html}

\date{\dateline{Feb 6, 2017}{Jan 8, 2018}{Jan 25, 2018}\\
\small Mathematics Subject Classifications: 05A20, 30C15}


\begin{document}

\maketitle



\begin{abstract}
  In the study of Kostka numbers and Catalan numbers, Kirillov posed a unimodality conjecture for the rectangular Narayana polynomials. We prove that the rectangular Narayana polynomials have only real zeros, and thereby confirm Kirillov's unimodality conjecture. By using an equidistribution property between descent numbers and ascent numbers on ballot paths due to Sulanke and a bijection between lattice words and standard Young tableaux, we show that the rectangular Narayana polynomial is equal to the descent generating function on standard Young tableaux of certain rectangular shape, up to a power of the indeterminate. Then we obtain the real-rootedness of the rectangular Narayana polynomial based on a result of Brenti which implies that the descent generating function of standard Young tableaux has only real zeros.  
  
\bigskip  \noindent \textbf{Keywords:} rectangular Narayana polynomials; lattice words;
  Young tableaux; Ferrers posets
\end{abstract}

\section{Introduction}

The main objective of this paper is to prove a unimodality conjecture for the
rectangular Narayana polynomials
in the study of Kostka numbers and Catalan numbers. This conjecture was first
posed by Kirillov \cite{Kirillov99} in 1999, and restated by himself
\cite{Kirillov15} in 2015. In this paper we prove that the rectangular
Narayana polynomials have only real zeros, an even stronger result than
Kirillov's conjecture.

Let us begin with an overview of Kirillov's conjecture. Throughout this paper, we abbreviate the vector
$(m,m,\ldots,m)$ with $n$ occurrences of $m$ as $(m^n)$ for any positive
integers $m$ and $n$.
We say that a word $\mathtt{w}=w_1w_2\cdots w_{nm}$ in symbols
$1,2,\ldots, m$ is a {lattice word} of weight $(m^n)$,
if the following
conditions hold:
\begin{itemize}
	\item[(a)] each $i$ between $1$ and $m$ occurs exactly $n$ times and
	
	\item[(b)] for each $1\leq r\leq nm$ and $1\leq i \leq m-1$, the number of
	$i$'s in $w_1w_2\cdots w_r$ is not less than the number of $(i+1)$'s.
\end{itemize}

Given a word $\mathtt{w}=w_1w_2\cdots w_p$ of length $p$, we say that $i$ is an
{ascent} of $\mathtt{w}$ if $w_i < w_{i+1}$, and a {descent} of
$\mathtt{w}$ if $w_i >w_{i+1}$. Denote the number of ascents of
$\mathtt{w}$ by  $\asc(\mathtt{w})$, and the number of descents
$\des(\mathtt{w})$. For any $m$
and $n$, the rectangular Narayana polynomial $N(n,m;t)$ is defined by
\begin{align}\label{def-Narayana_Pol}
N(n,m;t)= \sum_{\mathtt{w}\in \mathcal{N}(n,m)} t^{\des(\mathtt{w})},
\end{align}
where $\mathcal{N}(n,m)$ is the set of lattice words
of weight $(m^n)$. Note that $N(n,2;t)$ is the classical Narayana polynomial,
and
$N(n,2;1)$ is the classical Catalan number, see \cite{Kirillov15}. For this
reason,
$N(n,m;1)$ is called the rectangular Catalan number.

Kirillov's conjecture is concerned with the unimodality of the rectangular
Narayana polynomial $N(n,m;t)$.
Recall that a sequence $\{a_0,a_1,\ldots,a_n\}$ of positive real numbers is
said to be {unimodal} if there exists an integer $i\geq0$
such that
$$a_0\leq \cdots \leq a_{i-1}\leq a_i \geq a_{i+1}\geq\cdots\geq a_n,$$
and
{log-concave} if, for each $1\leq i\leq n-1$, there holds
$$a_i^2\geq a_{i-1}a_{i+1}.$$
Clearly, for a sequence of positive numbers, its log-concavity implies
unimodality.
Given a polynomial with real coefficients
$$f(t)=\sum_{k=0}^n a_k t^k,$$
it is unimodal (or log-concave) if its coefficient sequence
$\{a_0,a_1,\ldots,a_n\}$ is unimodal (resp. log-concave). Kirillov proposed the
following conjecture.

\begin{conj}[{\cite[Conjecture 2.5]{Kirillov15}}]
	\label{Conj-Kirillov}
	For any $m$ and $n$, the rectangular Narayana polynomial $N(n,m;t)$ is unimodal
	as a polynomial of $t$.
\end{conj}

In this paper, we give an affirmative answer to the above conjecture.
Instead of directly proving its unimodality, we shall show that the rectangular
Narayana polynomial $N(n,m;t)$ has only real zeros. By the well known Newton's inequality, if a polynomial with nonnegative coefficients has only
real zeros, then its coefficient sequence
must be log-concave and hence unimodal. Thus, from the real-rootedness of $N(n,m;t)$ we deduce its
log-concavity and unimodality.


The remainder of this paper is organized as follows.
In Section \ref{section-2}, we show that the rectangular Narayana polynomial
$N(n,m;t)$ is equal to the descent generating function on standard Young
tableaux of shape $(n^m)$, up to a power of $t$. We use a result of Sulanke
\cite{Sulanke04} that the ascent and descent statistics are equidistributed
over the set of ballot paths.
In Section \ref{section-3}, we first prove the real-rootedness of the descent
generating function on standard Young tableaux, and then obtain the
real-rootedness of $N(n,m;t)$.
The key to this approach is a connection between
the descent generating functions of standard Young tableaux and the Eulerian
polynomials of column-strict labeled Ferrers posets. The latter
polynomials have only real zeros, as proven by Brenti \cite{Brenti89} in the
study of the Neggers-Stanley conjecture.



\section{Tableau interpretation} \label{section-2}

The aim of this section is to interpret the rectangular
Narayana polynomials as the descent generating functions on standard
Young tableaux.

Let us first recall some definitions. Given an integer partition $\lambda=(\lambda_1, \lambda_2, \ldots, \lambda_{\ell})$, its
Young diagram is defined to be an array of squares in the plane
justified from the top left corner with $\ell$ rows and $\lambda_i$
squares in row $i$.  By transposing the diagram of $\lambda$, we get the
conjugate partition of $\lambda$, denoted $\lambda'$.
A cell $(i,j)$ of $\lambda$
is  in the $i$-th row from the top and in the $j$-th column from the left. A
{semistandard Young tableau (SSYT)} of shape $\lambda$ is a filling of its
diagram  by positive integers such
that it is weakly increasing in every row and strictly increasing down every
column. The type of $T$ is defined to be
the composition $\alpha=(\alpha_1,\alpha_2,\ldots)$, where $\alpha_i$ is the
number of $i$'s in $T$.
Let $|\lambda|=\lambda_1+\cdots+\lambda_{\ell}$.
If $T$ is of type $\alpha$ with $\alpha_i=1$ for $1\leq
i\leq |\lambda|$ and
$\alpha_i=0$ for $i> |\lambda|$, then it is called a {standard Young tableau
	(SYT)} of shape $\lambda$. Let $\mathcal{T}_{\lambda}$ denote the set of SYTs
of shape $\lambda$. Given a standard Young tableau,
we say that $i$ is a {descent} of $T$ if $i+1$ appears in a lower row of $T$
than $i$. Define the descent set $D(T)$ to be the set of all descents of $T$,
and denote by $\des(T)$ the
number of descents of $T$.

The main result of this section is as follows.

\begin{thm} \label{main-thm-0} For any positive integers $m$ and $n$, we have
	\begin{align}
	N(n,m;t)=t^{1-m}\sum_{T\in \mathcal{T}_{(n^m)}}t^{\des(T)}.
	\end{align}
\end{thm}

To prove the above result, we need a bijection between the set of lattice paths
and the set of standard Young tableaux. Here we use a very natural bijection
$\phi$ between the lattice word of weight $(m^n)$
and the
standard Young tableau of shape $(n^m)$,
see \cite[p. 92]{Coleman68}, \cite[p. 221]{Hamermesh89} and \cite{Stanley99}.
To be self-contained, we shall give a description of
this bijection in the following.

Given a lattice word $\mathtt{w}=w_1\cdots w_{nm}$ of weight $(m^n)$,  let
$T=\phi(\mathtt{w})$ be the tableau of shape $(n^m)$ obtained by filling
the  square $(i,j)$ with $k$ provided that $w_k$ is the $j$-th
occurrence of $i$ in $\mathtt{w}$ from left to right. Clearly,
$T$ is a standard Young tableau. Conversely, given a standard Young tableau $T$
of shape $(n^m)$, define a word $\mathtt{w}$ by letting $w_i$ to be
$j$ if $i$ is in the $j$-th row of $T$. It is easy to verify that
$\mathtt{w}=\phi^{-1}(T)$. Figure \ref{fig-1} gives an illustration of this
bijection, where $T$ is of shape $(4^3)$ and  $\mathtt{w}$ is of
weight $(3^4)$.

\begin{figure}[ht]
	\begin{center}
		$\mathtt{w}=~121113223233$~$\mapsto$~
		$T=$~
		\begin{ytableau}
			1 & 3 & 4 & 5 \\
			2 & 7 & 8 & 10\\
			6 & 9 & 11 & 12
		\end{ytableau}
	\end{center}
	\caption{Bijection between standard Young tableaux and lattice
		words}\label{fig-1}
\end{figure}

By using the above bijection $\phi$, we obtain the following result.

\begin{lem}
	For any positive integers $m$ and $n$, we have
	\begin{align}\label{D_TandA_w}
	\sum_{T\in \mathcal{T}_{(n^m)}}t^{\des(T)}=\sum_{\mathtt{w}\in\mathcal{N}(n,m)}
	t^{\asc(\mathtt{w})}.
	\end{align}
\end{lem}

\begin{proof}
	Suppose that $T=\phi(\mathtt{w})$.
	Note that if $i$ is an ascent in $\mathtt{w}$, i.e. $w_i< w_{i+1}$,
	then
	$i+1$ is filled in the $w_{i+1}$-th row, which is lower than
	the row including $i$ in $T$.
	Hence, $i$ is a descent of $T$.
	Conversely, given a tableau $T$, let $i$ be a descent of $T$ and  $\mathtt{w}=\phi^{-1}(T)$.
	Since $i+1$ appears in a lower row of $T$ than $i$, it follows that $w_{i}<w_{i+1}$.
	Hence, $i$ is an ascent of $\mathtt{w}$.
	Therefore, the bijection $\phi$ sends the set of ascents in $\mathtt{w}$ to the set of descents of $T=\phi(\mathtt{w})$ and hence   $\asc(\mathtt{w})=\des(T)$.
	This completes the proof.
\end{proof}
To prove Theorem \ref{main-thm-0}, it remains to show that
\begin{align}\label{eq-sun}
t^{1-m} \sum_{\mathtt{w}\in\mathcal{N}(n,m)}
t^{\asc(\mathtt{w})}=\sum_{\mathtt{w}\in\mathcal{N}(n,m)}
t^{\des(\mathtt{w})}.
\end{align}
In fact, this has been established by Sulanke \cite{Sulanke04}, who stated
it in terms of ballot paths. In the following, we
shall give an overview of Sulanke's result.

Recall that a ballot path for $m$-candidates is an
$m$-dimensional lattice path running from $(0,0,\ldots,0)$
to $(n,n,\ldots,n)$ with the steps:
\begin{align*}
X_1& =(1,0,\ldots,0),\\
X_2& =(0,1,\ldots,0),\\
\vdots &\qquad \qquad  \vdots \\
X_m& =(0,0,\ldots,1),
\end{align*}
and lying in the region
$$\{ (x_1,x_2,\ldots,x_m):0\leq x_1\leq x_2
\leq\cdots\leq x_m \}.$$ Denote by $\mathcal{C}(m,n)$ the set of all such paths.

For any path $P = p_1p_2\cdots p_{mn}\in \mathcal{C}(m,n)$,  {the number of
	ascents} of $P$ is defined by
$$\asc(P) = |\{i: p_i p_{i+1}=X_j X_l,j<l \}|,$$
and the number of descents of $P$ by
$$\des(P) = |\{i: p_i p_{i+1}=X_j X_l,j>l \}|.$$

Sulanke \cite{Sulanke04} obtained the following result by a nice bijection.

\begin{lem}[{\cite[Proposition 2]{Sulanke04}}]
	For any positive integers $m$ and $n$, we have
	\begin{align}
	\sum_{P\in \mathcal{C}(m,n)} t^{\asc(P)}=\sum_{P\in \mathcal{C}(m,n)}
	t^{\des(P)-m+1}.
	\end{align}
\end{lem}


Note that there is an obvious bijection between $\mathcal{C}(m,n)$ and
$\mathcal{N}(n,m)$: given a path $P\in\mathcal{C}(m,n)$, simply replace each
step $X_i$ of $P$ by the symbol $m-i+1$, and
the resulting word $\mathtt{w}$ is clearly a lattice word of
$\mathcal{N}(n,m)$. Moreover, we have $\asc(P)=\des(\mathtt{w})$ and
$\des(P)=\asc(\mathtt{w})$.
With this bijection, Sulanke's result can be restated as \eqref{eq-sun}.

\begin{proof}[Proof of Theorem \ref{main-thm-0}]
	Combining \eqref{def-Narayana_Pol}, \eqref{D_TandA_w} and \eqref{eq-sun},
	we immediately obtain the desired result.
\end{proof}

\section{Real zeros} \label{section-3}

In this section, we aim to prove the real-rootedness of rectangular Narayana
polynomials. Our main result of this section is as follows.

\begin{thm}\label{main-thm}
	The rectangular Narayana polynomial $N(n,m;t)$ has only real zeros for any $m$
	and $n$.
\end{thm}

By Theorem \ref{main-thm-0}, we only need to show that the following polynomial
$$\sum_{T\in \mathcal{T}_{(n^m)}}t^{\des(T)}$$
has only real zeros. To this end, we shall use a result due to Brenti \cite{Brenti89}
during his study of the Neggers-Stanley Conjecture. For more information on the Neggers-Stanley Conjecture, see \cite{Stembridge07} and references therein.
Given a partition $\lambda=(\lambda_1,\ldots,\lambda_{\ell})$,
the corresponding Ferrers poset $P_{\lambda}$
is the poset
\[ P_{\lambda}=\{(i,j)\in \mathbb{P} \times \mathbb{P}: 1\leq i\leq \ell, 1\leq
j\leq \lambda_i \},\]
ordered by the standard product ordering.
Let $\omega$ be a column strict labeling of
$P_{\lambda}$, namely,
$\omega(i,j)>\omega(i+1,j)$ and $\omega(i,j)<\omega(i,j+1)$ for all
$(i,j)\in P_{\lambda}$.
Brenti \cite{Brenti89} proved the following result, see also Br\"{a}nd\'{e}n
\cite{Branden04}.

\begin{thm}\cite[p. 60, Proof of Theorem 5.3.2]{Brenti89}
	Let $(P_{\lambda},\omega)$ be labeled column strict.
	Then the $(P_{\lambda},\omega)$\,-Eulerian polynomial
	\begin{align}
	W(P_{\lambda},\omega;t)=\sum_{\pi\in\mathcal{L}(P_{\lambda},\omega)} t^{\des(\pi)}
	\end{align}
	has only real zeros, where $\mathcal{L}(P_{\lambda},\omega)$ is the Jordan-H\"older set
	of $(P_{\lambda},\omega)$.
\end{thm}

Based on the above theorem, we obtain the following result.

\begin{coro}\label{WPrz} For any partition $\lambda$,
	the polynomial
	$$\sum_{T\in \mathcal{T}_{\lambda}}t^{\des(T)}$$
	has only real zeros.
\end{coro}

\begin{proof}
	It suffices to show that
	\begin{align}
	W(P_{\lambda},\omega;t)=\sum_{T\in \mathcal{T}_{\lambda}}t^{\des(T)}.
	\end{align}
	To this end, we need to establish a bijection $\psi$ from
	$\mathcal{L}(P_{\lambda},\omega)$ to $\mathcal{T}_{\lambda}$ such that
	$\des(\pi)=\des(\psi(\pi))$ for any $\pi\in \mathcal{L}(P_{\lambda},\omega)$. Suppose that $|\lambda|=p$.
	Note that, given a permutation
	$\pi=\pi_1\pi_2\cdots\pi_p\in \mathcal{L}(P_{\lambda},\omega)$, the sequence
	$\omega^{-1}(\pi_1)\omega^{-1}(\pi_2)\cdots\omega^{-1}(\pi_p)$ is a linear
	extension of $P_{\lambda}$.
	Let $\psi(\pi)$ be the tableau of shape $\lambda$ by filling
	the square $\omega^{-1}(\pi_k)$ with $k$. Since $\omega$ is column strict, it is readily to see that $\psi(\pi)$ is a standard Young tableau and $\psi$ is a bijection. Furthermore,  $k$ is a descent in $\pi$ if and only if
	$k$ is a descent in $\psi(\pi)$. In fact, suppose that $k$ and $k+1$ are in the square
	$(x,y)$ and $(x',y')$ then $\pi_k>\pi_{k+1}$ implies that $x<x'$, that is,
	$k+1$ appears in a lower row of $\psi(\pi)$ than $k$.
	For example, taking $\pi=4215673$ and the labeling $\omega$ showing in the
	Figure \ref{perm-tab}, we obtain the standard Young tableau $\psi(\pi)$.
\end{proof}

\begin{figure}[ht]
	\begin{center}
		$\omega=$~
		\begin{ytableau}
			4 & 5 & 6 & 7 \\
			2 & 3 \\
			1
		\end{ytableau}~,
		~~~
		$\pi=4215673$~$\mapsto$~
		$\psi(\pi)=$~
		\begin{ytableau}
			1 & 4 & 5 & 6 \\
			2 & 7 \\
			3
		\end{ytableau}
	\end{center}
	\caption{Bijection between permutations in $\mathcal{L}(P_{\lambda},\omega)$ and standard
		Young tableaux of shape $\lambda$ for a given labeling $\omega$.}\label{perm-tab}
\end{figure}

Now we can give a proof of Theorem \ref{main-thm}.

\noindent \textit{Proof of Theorem \ref{main-thm}.} This follows
from Theorem
\ref{main-thm-0} and Corollary \ref{WPrz}.

As an immediate corollary of Theorem \ref{main-thm}, we obtain the following result, which gives
an affirmative answer to Kirillov's conjecture.

\begin{coro}
	The rectangular Narayana polynomial $N(n,m;t)$ is unimodal for any $m$ and $n$.
\end{coro}

\subsection*{Acknowledgements} We would like to thank the anonymous referee for the valuable comments. 
%This work was supported by the National Science Foundation of China (Nos. 11231004, 11522110, 11626172, 11701424).
%The first author  was partially supported by the NSFC grant (11701492) and the PHD Program  of Tianjin Chengjian University (160501436).
%The third author  was partially supported by the PHD Program  of Tianjin Normal University (52XB1616).

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\end{thebibliography}


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