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\title{\bf A group action on derangements\thanks{Supported by  the National Natural Science Foundation of China (Nos. 11071030, 11371078) and the Specialized Research Fund for the Doctoral Program of Higher
Education of China (No. 20110041110039).}}

% input author, affilliation, address and support information as follows;
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% the street address

\author{Hua Sun\\
\small School of Mathematical Sciences\\[-0.8ex]
\small Dalian University of Technology\\[-0.8ex]
\small Dalian 116024, PR China\\
\small\tt hanch.sun@gmail.com\\
\and
Yi Wang\\
\small School of Mathematical Sciences\\[-0.8ex]
\small Dalian University of Technology\\[-0.8ex]
\small Dalian 116024, PR China\\
\small\tt wangyi@dlut.edu.cn
}

% \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{Sep 18, 2013}{Mar 14, 2014}{Mar 24, 2014}\\
\small Mathematics Subject Classifications: 05A15, 05A05, 05E18, 26C99}

\begin{document}

\maketitle

% E-JC papers must include an abstract. The abstract should consist of a
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\begin{abstract}
  In this paper we define a cyclic analogue of the MFS-action on derangements,
  and give a combinatorial interpretation of the expansion
  of the $n$-th derangement polynomial on the basis $\{q^{k}(1+q)^{n-1-2k}\},k=0,1,\ldots,\lrf{(n-1)/2}$.

  % keywords are optional
  \bigskip\noindent \textbf{Keywords:} derangement polynomials; group action
\end{abstract}

\section{Introduction}

Let $[n]$ denote the set $\{1,2,\ldots,n\}$ and let $\cs_{n}$ denote the set of all permutations of $[n]$.
For $\pi=\pi_{1}\pi_{2}\cdots\pi_{n}\in\cs_{n}$ and $x\in [n]$,
we write $\pi$ as the concatenation $\pi=w_{1}w_{2}xw_{3}w_{4}$,
where $w_{2}$ is the maximal contiguous subword immediately to the
left of $x$ whose letters are all smaller than $x$, and $w_{3}$ is
the maximal contiguous subword immediately to the right of $x$ whose
letters are all smaller than $x$. Following Foata and Strehl~\cite{FS74,FS76},
this concatenation is called the {\it $x$-factorization} of $\pi$. For example, let $\pi=714358296$
and $x=5$. Then $w_{1}=7$, $w_{2}=143$, $w_{3}=\emptyset$ and
$w_{4}=8296$.

Foata and Strehl~\cite{FS74,FS76} defined an involution acting on $\cs_{n}$ by
$\varphi_{x}(\pi)=w_{1}w_{3}xw_{2}w_{4}$ for $x\in[n]$ and $\varphi_{S}(\pi)=\prod_{x\in S}\varphi_{x}(\pi)$
for $S\subseteq [n]$. The group
$\mathbb{Z}_{2}^{n}$ acts on $\cs_{n}$ via the functions
$\varphi_{S}$ for $S\subseteq [n]$.

\begin{definition}\label{DEF1}
Let $\pi=\pi_{1}\pi_{2}\cdots \pi_{n}\in\cs_{n}$ and denote $\pi_{0}=\pi_{n+1}=n+1$.
The entry $\pi_{k}$ is called a {\it valley} if $\pi_{k-1}>\pi_{k}<\pi_{k+1}$;
a {\it peak} if $\pi_{k-1}<\pi_{k}>\pi_{k+1}$;
a {\it double ascent} if $\pi_{k-1}<\pi_{k}<\pi_{k+1}$;
a {\it double descent} if $\pi_{k-1}>\pi_{k}>\pi_{k+1}$.
\end{definition}

Let $Val(\pi)$, $Peak(\pi)$, $Dasc(\pi)$, $Ddes(\pi)$ denote the set of all valley, peaks, double ascents and
double descents of $\pi$, respectively.
The corresponding cardinalities are $val(\pi)$, $peak(\pi)$, $dasc(\pi)$ and $ddes(\pi)$, respectively.
Shapiro {\it et al}.~\cite{SWG83} modified the Foata-Strehl action
in the following way. For $x\in[n]$, let
\begin{equation}
\varphi_{x}'(\pi)=
\begin{cases}
\varphi_{x}(\pi) & \text{if $x$ is a double ascent or a double descent,}\\
\pi & \text{if $x$ is a valley or a peak.}
\end{cases}
\end{equation}
For any subset $S\subseteq[n]$, define $\varphi_{S}'(\pi)=\prod_{x\in S}\varphi_{x}'(\pi)$.
From the definition, if $x$ is a double
ascent (double descent, resp.) of $\pi$, then $x$ is a double
descent (double ascent, resp.) of $\varphi_{x}'(\pi)$. The group
$\mathbb{Z}_{2}^{n}$ acts on $\cs_{n}$ via the functions
$\varphi_{S}',S\in [n]$ and call this action the {\it MFS-action}.

By the theory of symmetric functions, Brenti~\cite{Bre90} showed that derangement polynomials are symmetric and unimodal polynomials.
Using the method of continued fractions, Shin and Zeng~\cite{SZ12} gave a combinatorial interpretation for  coefficients in the expansion of the $n$-th derangement polynomial on the basis
$\{q^{k}(1+q)^{n-1-2k}\},k=0,1,\ldots,\lrf{(n-1)/2}$. In this note,
we define a cyclic analogous of the MFS-action on derangements and give a new proof for the result of Shin and Zeng.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Main results}

Let $\pi\in\cs_{n}$. We say that $\pi$ is a {\it derangement} of $[n]$ if $\pi_{i}\neq i$ for all $i\in[n]$.
Denote by $D_{n}$ the set of all derangements of $[n]$. An element $i\in[n]$ is an {\it excedance} of $\pi$ if $\pi_{i}>i$.
Denote by $Exc(\pi)$ the set of all excedances in $\pi$ and let $exc(\pi)=|Exc(\pi)|$.
The {\it $n$-derangement polynomial} $D_{n}(q)$ is the generating function of
statistic excedance over the set $D_{n}$, i.e.,
\begin{equation}
D_{n}(q)=\sum_{\pi\in
D_{n}}q^{exc(\pi)}=\sum_{j=1}^{n-1}d(n,j)q^j,
\end{equation}
where $d(n,j)=|\{\pi\in D_{n}: exc(\pi)=j\}|.$

Recall that a permutation $\pi\in \cs_{n}$ may be regarded as a
disjoint union of its distinct cycles $C_{1},C_{2},\ldots,C_{k}$,
written $\pi=C_{1}C_{2}\cdots C_{k}$. Let $c(\pi)$ denote the number of cycles of $\pi$.
For a derangement $\pi$, each cycle contains at least two elements.
The {\it standard cycle representation} of $\pi$ is defined by requiring that (\romannumeral1) each cycle is
written with its largest element first, and (\romannumeral2) the cycles are
written in increasing order of their largest elements~\cite{Sta97}.
For example, the standard cycle representation of $\pi=456321\in
D_{6}$ is $(52)(6143)$. Throughout the paper all permutations are written in standard cycle representation.

\begin{definition}[\cite{SZ12}]\label{DEF2}
Let $\pi\in\cs_{n}$. The entry $x=\pi_{i}(i\in[n])$ is called
a {\it cyclic valley} if $i=\pi^{-1}(x)>x<\pi(x)$;
a {\it cyclic peak} if $i=\pi^{-1}(x)<x>\pi(x)$;
a {\it cyclic double ascent} if $i=\pi^{-1}(x)<x<\pi(x)$;
a {\it cyclic double descent} if $i=\pi^{-1}(x)>x>\pi(x)$;
a {\it fixed point} if $\pi(x)=x$.
\end{definition}

Let $Cval(\pi)$, $Cpeak(\pi)$, $Cdasc(\pi)$, $Cddes(\pi)$ and $Fix(\pi)$ denote the set of all cyclic valley, cyclic peaks, cyclic double ascents, cyclic double descents and fixed points of $\pi$, respectively. The corresponding cardinalities are $cval(\pi)$, $cpeak(\pi)$, $cdasc(\pi)$, $cddes(\pi)$ and $fix(\pi)$, respectively.
It is easy to see that the union of sets $Cval(\pi)$, $Cpeak(\pi)$, $Cdasc(\pi)$,
$Cddes(\pi)$ and $Fix(\pi)$ is $[n]$ for any $\pi\in\cs_{n}$.
For a derangement $\pi$, the set $Fix(\pi)$ is empty.
The following proposition is immediate by Definition~\ref{DEF2}.

\begin{proposition}\label{Dn-exc}
Let $\pi=C_{1}C_{2}\cdots C_{k}$ be a permutation of $[n]$. Then
$$Exc(\pi)=Cval(\pi)\cup Cdasc(\pi)$$
and $$ exc(\pi)=cval(\pi)+cdasc(\pi).$$
\end{proposition}

Let $\pi=C_{1}C_{2}\cdots C_{k}$.
Following Stanley~\cite{Sta97},
let $o(\pi)$ be the permutation obtained from $\pi$ by erasing the parentheses of cycles.
For example, if $\pi=(71435)(826)$, then $o(\pi)=71435862$.
The map $o:\cs_{n}\rightarrow\cs_{n}$ defined above is a bijection.
The following result is direct.

\begin{proposition}\label{c-nonc}
Let $\pi=C_{1}C_{2}\cdots C_{k}\in D_{n}$. Suppose that $o(\pi)(0)=0$ and $o(\pi)(n+1)=n+1$. Then
$$Cpeak(\pi)=Peak(o(\pi)),\qquad Cval(\pi)=Val(o(\pi)),$$
$$Cdasc(\pi)=Dasc(o(\pi))\qquad \text{and}\qquad Cddes(\pi)=Ddes(o(\pi)),$$
where the sets $Peak(o(\pi)),Val(o(\pi)),Dasc(o(\pi))$ and
$Ddes(o(\pi))$ are defined similar to Definition~\ref{DEF1}
with the only difference $o(\pi)(0)=0$.
\end{proposition}

We define the cyclic analogous of the MFS-action on derangements in
the following way. Let $\pi=C_{1}C_{2}\cdots C_{k}$. Suppose that $o(\pi)(0)=0$ and $o(\pi)(n+1)=n+1$.
For $x\in[n]$, define the map $\theta_{x}:D_{n}\rightarrow D_{n}$
by
$$\theta_{x}(\pi)=o^{-1}(\varphi_{x}'(o(\pi))).$$

The map is well-defined. To see this, let $\pi=C_{1}C_{2}\cdots C_{k}\in D_{n}$.
If $x$ is a cyclic valley of $\pi$, then $x$ is a valley of $o(\pi)$,
$\varphi_{x}'(o(\pi))=o(\pi)$ and $\theta_{x}(\pi)=\pi$.
If $x$ is a cyclic peak of $\pi$, then $x$ is a peak of $o(\pi)$,
$\varphi_{x}'(o(\pi))=o(\pi)$ and $\theta_{x}(\pi)=\pi$.
If $x$ is a cyclic double ascent of $C_{i}$ in $\pi$,
where $C_{i}=(w_{0}w_{1}xw_{2})$ and $w_{1}$ denotes the maximal contiguous subword immediately to the
left of $x$ whose letters are all smaller than $x$.
Then $x$ is a double ascent of $o(\pi)$,
$\varphi_{x}'(o(\pi))=o(C_{1}C_{2}\cdots C_{i-1}\bar{C_{i}}C_{i+1}\cdots C_{k})$ and $\theta_{x}(\pi)=C_{1}C_{2}\cdots C_{i-1}\bar{C_{i}}C_{i+1}\cdots C_{k}\in D_{n}$,
where  $\bar{C_{i}}=(w_{0}xw_{1}w_{2})$.
If $x$ is a cyclic double descent of $C_{i}$ in $\pi$,
where $C_{i}=(w_{0}xw_{1}w_{2})$ and $w_{1}$ denotes the maximal contiguous subword immediately to the
right of $x$ whose letters are all smaller than $x$.
Then $x$ is a double descent of $o(\pi)$,
$\varphi_{x}'(o(\pi))=o(C_{1}C_{2}\cdots C_{i-1}\bar{C_{i}}C_{i+1}\cdots C_{k})$ and $\theta_{x}(\pi)=C_{1}C_{2}\cdots C_{i-1}\bar{C_{i}}C_{i+1}\cdots C_{k}\in D_{n}$,
where $\bar{C_{i}}=(w_{0}w_{1}xw_{2})$.
Hence the map $\theta_{x}$ is well-defined for all $x\in [n]$.

Table~1 gives an example of the maps $\theta_{x}$ on $\pi=(623)(87514)$ for all $x\in[8]$, where $o(\pi)=62387514$.

\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
$x$ & 1 & 2 & 3 & 4 \\
\hline
$\varphi'_x(o(\pi))$ & 62387514 & 62387514 & 63287514 & 62387514 \\
\hline
$\theta_x(\pi)$ & (623)(87514) & (623)(87514) & (632)(87514) &
   (623)(87514) \\
\hline
\hline
$x$ &  5 & 6 & 7 & 8 \\
\hline
$\varphi'_x(o(\pi))$ & 62387145 & 62387514 & 62385147 & 62387514\\
\hline
$\theta_x(\pi)$ & (623)(87145) & (623)(87514) & (623)(85147) & 
        (623)(87514)\\
\hline
\end{tabular}
\medskip
\end{center}
\centerline{Table 1.}

\bigskip

%% CSG, 17.3.2014: reformatted table as above.

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\put(1.7,1.3){$62387514$}\put(3.4,1.3){$62387514$}\put(5.1,1.3){$63287514$}\put(6.8,1.3){$62387514$}
%\put(8.5,1.3){$62387145$}\put(10.2,1.3){$62387514$}\put(11.9,1.3){$62385147$}\put(13.6,1.3){$62387514$}
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%\put(12.1,0.2){$(81457)$}\put(12.3,0.6){$(623)$}\put(13.8,0.2){$(87514)$}\put(14,0.6){$(623)$}
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%\put(7,-0.8){Table~1. }
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%\hspace{4mm}
%
%~~~~~

The function $\theta_{x}$ is an involution and 
$\theta_{x}\theta_{y}=\theta_{y}\theta_{x}$ for all $x,y\in[n]$.
For any subset $S\subseteq[n]$, define the
function $\theta_{S}(\pi):D_{n}\rightarrow D_{n}$ by
$$\theta_{S}(\pi)=\prod_{x\in S}\theta_{x}(\pi).$$
The group $\mathbb{Z}_{2}^{n}$ acts on $D_{n}$ via the functions
$\theta_{S},S\in [n]$ and call this action the {\it CMFS-action}.

For $\pi\in D_{n}$, let $Orb^{c}(\pi)$ denote the orbit including $\pi$ under the CMFS-action. There is a unique derangement in $Orb^{c}(\pi)$, denoted by $\tilde{\pi}$, such that $\tilde{\pi}$ has no cyclic double ascents.
The next is the main results of this note.

\begin{theorem}\label{Derangement-orbit}
Let $\pi\in D_{n}$. Then
$$\sum_{\sigma\in Orb^{c}(\pi)}q^{exc(\sigma)}
=q^{exc(\tilde{\pi})}(1+q)^{n-2exc(\tilde{\pi})}=q^{cpeak(\pi)}(1+q)^{n-2cpeak(\pi)}.$$
\end{theorem}

\begin{proof}
If $x$ is a cyclic double descent of some cycle $C_{i}$ in $\pi$,
then $x$ is a cyclic double ascent of cycle $C'_{i}$ in $\theta_{x}(\pi)$,
where $\pi=C_{1}C_{2}\cdots C_{k}$ and $\theta_{x}(\pi)=C'_{1}C'_{2}\cdots C'_{k}$.
We have $Cdasc(\theta_{x}(\pi))=Cdasc(\pi)\cup\{x\}$ and $Cval(\theta_{x}(\pi))=Cval(\pi)$.
It follows that $Exc(\theta_{x}(\pi))=Exc(\pi)\cup\{x\}$ and $exc(\theta_{x}(\pi))=exc(\pi)+1$ from Proposition~\ref{Dn-exc}.
Then
$$\sum_{\sigma\in Orb^{c}(\pi)}q^{exc(\sigma)}=q^{exc(\tilde{\pi})}(1+q)^{cddes(\tilde{\pi})}.$$

For any $\pi=C_{1}C_{2}\cdots C_{k}\in D_{n}$,
delete all double descents and double ascents of $o(\pi)$,
then we get an alternating permutation
$$0<x_{1}>x_{2}<x_{3}>\cdots>x_{n-cddes(\pi)-cdasc(\pi)}<n+1,$$
where $o(\pi)(0)=0$ and $o(\pi)(n+1)=n+1$.
Thus
$$cpeak(\pi)=peak(o(\pi))=val(o(\pi))=cval(\pi).$$
Note that the union of sets $Cval(\tilde{\pi})$, $Cpeak(\tilde{\pi})$ and $Cddes(\tilde{\pi})$ is the set $[n]$.
Hence $exc(\tilde{\pi})=cpeak(\tilde{\pi})=cpeak(\pi)$ and $cddes(\tilde{\pi})=n-2exc(\tilde{\pi})=n-2cpeak(\pi)$.
\end{proof}

The following corollary is an immediate consequence of Theorem~\ref{Derangement-orbit}.

\begin{corollary}[\cite{SZ12}]\label{Derangement}
The derangement polynomials can be expanded as
$$D_{n}(q)=\sum_{i=0}^{\lrf{n/2}}b_{i}q^{i}(1+q)^{n-2i},$$
where $b_{i}=2^{-n+2i}|\{\pi\in D_{n}:cpeak(\pi)=i\}|$ and $b_{0}=0$.
\end{corollary}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Acknowledgements}
The authors thank the anonymous referee for his/her careful reading and helpful comments.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \bibliographystyle{plain}
% \bibliography{myBibFile}
% If you use BibTeX to create a bibliography
% then copy and past the contents of your .bbl file into your .tex file

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\end{thebibliography}
\end{document}
