<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">%% if you are submitting an initial manuscript then you should have submission as an option here
%% if you are submitting a revised manuscript then you should have revision as an option here
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\documentclass[finalversion]{FPSAC2020}
\articlenumber{40}
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%% note that the class file already loads {amsmath, amsthm, amssymb}

\theoremstyle{plain}
\newtheorem{thm}{Theorem}[section]
\newtheorem{lem}[thm]{Lemma}
\newtheorem{cor}[thm]{Corollary}
\theoremstyle{definition}
\newtheorem{remark}[thm]{Remark}
\newtheorem{example}[thm]{Example}


\usepackage{tikz}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\N{\mathbb{N}}
\def\Z{\mathbb{Z}}
\def\S{\mathfrak{S}}
\def\D{\mathfrak{D}}
\def\L{\mathcal{LH}}
\def\A{\mathfrak{A}}
\def\M{\mathcal{M}}
\def\O{\mathcal{O}}
\def\R{\mathbb{R}}
\def\B{\mathcal{B}}

%%%%


\def\ai{\mathsf{ai}}

\def\valley{\mathsf{val}}
\def\val{\mathsf{val}}
\def\peak{\mathsf{pk}}
\def\da{\mathsf{da}}
\def\dd{\mathsf{dd}}
%%%
\def\cvalley{\mathsf{cval}}
\def\cval{\mathsf{cval}}
\def\cpeak{\mathsf{cpk}}
\def\cpk{\mathsf{cpk}}
\def\cda{\mathsf{cda}}
\def\cdd{\mathsf{cdd}}

%%
\def\lpeak{\mathsf{lpk}}
\def\lvalley{\mathsf{lval}}
\def\ldd{\mathsf{ldd}}
\def\lda{\mathsf{lda}}

%%
\def\rpeak{\mathsf{rpk}}
\def\rvalley{\mathsf{rval}}
\def\rda{\mathsf{rda}}
\def\rdd{\mathsf{rdd}}


%%
\def\sde{\mathsf{scda}}
\def\sdn{\mathsf{scdn}}
\def\scpeak{\mathsf{scpk}}
\def\scval{\mathsf{scval}}

%%%
\def\des{\mathsf{des}}
\def\asc{\mathsf{asc}}
\def\exc{\mathsf{exc}}
\def\drop{\mathsf{drop}}
\def\wex{\mathsf{wex}}
\def\inv{\mathsf{inv}}
\def\fix{\mathsf{fix}}
\def\cros{\mathsf{cros}}
\def\nest{\mathsf{nest}}
\def\icr{\mathsf{icr}}
\def\ine{\mathsf{ine}}
\def\cab{31\text{-}2}
\def\bca{2\text{-}31}
\def\acb{13\text{-}2}
\def\bac{2\text{-}13}
%%%%



\def\cdes{\mathsf{cdes}}
\def\fdes{\mathsf{fdes}}
\def\fexc{\mathsf{fexc}}
\def\wexa{\mathsf{wex_A}}
\def\exca{\mathsf{exc_A}}
\def\wexc{\mathsf{wex_C}}
\def\ndes{\mathsf{ndes}}
\def\fmax{\mathsf{fmax}}
\def\neg{\mathsf{neg}}
\def\Orb{\mathsf{Orb}}
\def\st{\mathsf{st}}
\def\stat{\mathsf{stat}}
\def\cyc{\mathsf{cyc}}
\def\pos{\mathsf{pos}}
\def\fmin{\mathsf{fmin}}
\def\amax{\mathsf{amax}}
\def\amin{\mathsf{amin}}
%%
\def\fmax{\mathsf{fmax}}
\def\amax{\mathsf{amax}}
\def\area{\mathsf{area}}
\def\MAD{\mathsf{MAD}}



\DeclareMathOperator\Cval{\mathrm{Cval}}
\DeclareMathOperator\Cpeak{\mathrm{Cpk}}
\DeclareMathOperator\Cda{\mathrm{Cda}}
\DeclareMathOperator\Fix{\mathrm{Fix}}
\DeclareMathOperator\Cdd{\mathrm{Cdd}}
\def\DD{\mathrm{DD}}
\def\DE{\mathrm{DE}}
\def\SDE{\mathrm{SDE}}

\def\Ine{\mathsf{Ine}}
\def\Icr{\mathsf{Icr}}
 %%%%%%%%%%
 \DeclareMathOperator\EB{EB}
\DeclareMathOperator\SE{SE}
\DeclareMathOperator\NE{NE}
\DeclareMathOperator\OSE{OSE}
\DeclareMathOperator\ENE{ENE}
\DeclareMathOperator\ER{ER}

%%%%%%%%%%%%%%%
 
 \def\Dyck{\mathsf{Dyck}}
 \def\rank{\mathsf{rank}}
 \def\Av{\mathsf{Av}}
 \def\LH{\mathcal{LH}}
 \def\su{\textsc{U}}
 \def\sd{\textsc{D}}
 \def\lr{\textsc{L}_r}
 \def\la{\textsc{L}_r}
 \def\lb{ \textsc{L}_b}
 \def\lc{ \textsc{L}_c}
 \def\Drop{\mathsf{Drop}}
 \def\Asc{\mathsf{Asc}}
 
 %%%%%%%%%%%%%
 \def\Wex{\mathrm{Wex}}
\def\Ine{\mathsf{Ine}}
\def\Icr{\mathsf{Icr}}
\def\Lval{\mathsf{Lval}}
\def\Lpeak{\mathsf{Lpk}}
\def\Lda{\mathsf{Lda}}
\def\Ldd{\mathsf{Ldd}}
\def\Val{\mathsf{Val}}
\def\Peak{\mathsf{Pk}}
\def\Da{\mathsf{Da}}
\def\Dd{\mathsf{Dd}}
\def\Sval{\mathsf{Sval}}
\def\Speak{\mathsf{Spk}}
\def\Sda{\mathsf{Sda}}
%\def\Sdn{\mathsf{Sdn}} 
 

%%%%%%%%%%%%%%
\DeclareMathOperator\Scval{\mathrm{Scval}}
%\DeclareMathOperator\Cval{\mathrm{Cval}}
\DeclareMathOperator\Scpeak{\mathrm{Scpeak}}
%\DeclareMathOperator\Cpeak{\mathrm{Cpeak}}
\DeclareMathOperator\Sde{\mathrm{Scda}}
\DeclareMathOperator\Sdn{\mathrm{Scdn}}
\def\Sne{\mathsf{Snest}}
\def\Scr{\mathsf{Scros}}\DeclareMathOperator\Nest{\mathrm{Nest}}\DeclareMathOperator\Cros{\mathrm{Cros}}

\DeclareMathOperator\Recp{\mathrm{Recp}}
\DeclareMathOperator\Arecp{\mathrm{Arecp}}
\DeclareMathOperator\Arec{\mathrm{Arec}}
\DeclareMathOperator\Rec{\mathrm{Rec}}
\DeclareMathOperator\Exc{\mathrm{Exc}}
\DeclareMathOperator\Erec{\mathrm{Erec}}
\DeclareMathOperator\Recl{\mathrm{Recl}}
\DeclareMathOperator\Arecl{\mathrm{Arecl}}
\DeclareMathOperator\Cyc{\mathrm{Cyc}}
\DeclareMathOperator\Rar{\mathrm{Rar}}
\DeclareMathOperator\Rarp{\mathrm{Rarp}}
\DeclareMathOperator\Rarl{\mathrm{Rarl}}




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%% define your title in the usual way
\title[Eulerian polynomials and excedance statistics via continued fractions]{Eulerian polynomials and excedance statistics via continued fractions}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{Bin Han\thanks{\href{mailto:han@math.biu.ac.il}{han@math.biu.ac.il}. 
Partially supported by the Israel Science Foundation, grant no. 1970/18 and by the China Scholarship Council.}\addressmark{1}, 
Jianxi Mao\thanks{\href{mailto:maojianxi@hotmail.com}{maojianxi@hotmail.com}. Supported by the China Scholarship Council.}\addressmark{2}, \and Jiang Zeng \thanks{\href{mailto:zeng@math.univ-lyon1.fr}{zeng@math.univ-lyon1.fr}.}\addressmark{3}}

%% then use \addressmark to match authors to institutions here
\address{
\addressmark{1}Department of Mathematics, Bar-Ilan University, Ramat-Gan, Israel. \\
\addressmark{2}School of Mathematic Sciences, Dalian University of Technology, Dalian, China\\
\addressmark{3}Univ Lyon, Universit\'e Claude Bernard Lyon 1, CNRS, Institut Camille Jordan, Villeurbanne, France}

%% put the date of submission here
\received{\today}

%% leave this blank until submitting a revised version
%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{It is well known that 
the permutation peak polynomials and descent  polynomials are connected 
via a quadratic transformation. By rephrasing  the latter  formula  with
 permutation cycle peaks and excedances 
we are able to prove a series of general formulas expressing polynomials counting permutations by various excedance statistics in terms of refined Eulerian polynomials.}  

%% put your French abstract here, or comment this out if you don't have one
\resume{Il est bien connu que les polyn\^omes de pic de permutations et les polyn\^omes de descente sont connect\'es via une transformation quadratic. En reformulant cette derni\`ere formule avec 
les pics de cycle et les exc\'edances du cycle de permutation, nous pouvons prouver une s\'erie de formules g\'en\'erales exprimant des polyn\^omes \'enum\'eratives des permutations par diverses statistiques d'exc\'edance en termes de polyn\^omes eul\'eriens raffin\'es.}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{Eulerian polynomials, peak polynomials, Gamma-positivity,  $q$-Narayana polynomials, continued fractions, Laguerre histories, Fran\c con--Viennot bijection, Foata--Zeilberger bijection.}

%% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easier
%% regardless of how you do it, please use numerical citations, ie. [xx, yy] in the text

%% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex
%% but you can use bibtex if you want
\addbibresource{Han-Mao-Zeng.bib}
%% note the \printbibliography command at the end of the file which goes with these biblatex commands

\begin{document}

\maketitle
%% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above

\section{Introduction}

The Eulerian polynomials $A_n(t)$ can be defined 
through the continued fraction expansion~\cite{Sti89}
\begin{align}\label{gf:eulerian}
%\sum_{n=0}^{\infty}A_{n}(t)\frac{x^{n}}{n!}=\frac{t-1}{t-e^{(t-1)x}}.
 \sum_{n\geq 0}A_n(t)z^n=1/1-1\cdot z/1-t\cdot z/1-2\cdot z /1-2t\cdot z/1-3\cdot z/1-3t\cdot z/1-\ldots
   \end{align}
For  an $n$-permutation 
$\sigma:=\sigma(1)\sigma(2)\cdots\sigma(n)$ of the word $1\ldots n$,
an index  $i$ ($1\leq i\leq n-1$) is a \textit{descent} (resp. \textit{excedance}) of $\sigma$
 if $\sigma(i)&gt;\sigma(i+1)$ (resp. $\sigma(i)&gt;i$).   
 It is well-known~\cite{FS70, Pet15} that   
\begin{align}\label{des-exc}
A_{n}(t)=\sum_{\sigma\in\mathfrak{S}_{n}}t^{\des\, \sigma}=\sum_{\sigma\in\mathfrak{S}_{n}}t^{\exc\,\sigma},
\end{align}
where $\mathfrak{S}_{n}$ is  the set of $n$-permutations and 
$\des\,\sigma$ (resp. $\exc\,\sigma$) denotes the number of descents (resp. excedances) of $\sigma$.
The value  $\sigma(i)$ ($2\leq i\leq n-1$) is a \textit{peak} of
$\sigma$ if $\sigma(i-1)&lt;\sigma(i)&gt;\sigma(i+1)$ and 
the \emph{peak polynomials} are defined by 
\begin{equation}\label{eq:peak-poly}
P_{n}^{\peak}(x):=\sum_{\sigma\in\mathfrak{S}_{n}}x^{\peak'\,\sigma}
\end{equation}
where  $\peak'\,\sigma$ denotes the number of peaks of $\sigma$.
The peak polynomials are related to the Eulerian polynomials 
by Stembridge's identity, %~\cite[Remark 4.8]{Ste97},   
see \cite{Bra08, Zhuang17}.
\begin{equation}\label{eq:stembridge}
A_{n}(t)=\left(\frac{1+t}{2}\right)^{n-1}P_{n}^{\peak}\left(\frac{4t}{(1+t)^{2}}\right),
\end{equation}
which can be used  to compute the peak polynomials.
Obviously \eqref{eq:stembridge}  is equivalent to the so-called $\gamma$-expansion of Eulerian polynomials
\begin{equation}\label{FSGP}
A_n(t)=\sum_{k=0}^{\lfloor (n-1)/2\rfloor} 2^{2k+1-n}\gamma_{n,k}t^k(1+t)^{n-1-2k},
\end{equation}
where $\gamma_{n,k}$ is the number of $n$-permutations with $k$ peaks.
In  the form of \eqref{FSGP} it is not difficult to see that 
Stembridge's formula~\eqref{eq:stembridge}  is actually equivalent to 
a formula of Foata and Sch\"{u}zenberger~\cite[Th\'eor\`eme 5.6]{FS70} via Br\"{a}nd\'{e}n's modified Foata-Strehl action (cf. ~\cite{Bra08}).
In the last two decades, 
many people studied the refinements of Stembridge's identity, 
see
Br\"{a}nd\'{e}n~\cite{Bra08}, 
Shin and Zeng~\cite{SZ12, SZ16}, Zhuang~\cite{Zhuang17},
Athanasiadis~\cite{Ath18} and the references therein.
In particular, Zhuang~\cite{Zhuang17} has proved 
several identities expressing polynomials counting permutations by various descent statistics in terms of Eulerian polynomials, extending results of Stembridge, Petersen, and Br\"and\'en.   


In this paper we shall   prove generalizations of 
Stembridge's formula using  excedance  statistics 
by further exploiting the continued fraction technique in~\cite{SZ12, SZ16}.
Our main tool is the combinatorial theory  of continued fractions due to
Flajolet~\cite{Fla80} and  bijections due to Fran\c{c}on-Viennot, Foata--Zeilberger between
permutations and \emph{Laguarre histories}, see \cite{FV79, FZ90, Fla80, CSZ97}. As in \cite{SZ12} this approach  uses both linear and cycle statistics on permutations. 

This extended abstract is a summary of the recent paper \cite{HMZ19}. In \cref{sec:2} we introduce the work in \cite{SZ12,SZ16}, and construct a
bijection $\psi$, which is an analogue  
of Foata--Zeilberger's bijection from $\S_{n+1}$ to $\LH_n$. 
In \cref{sec:3} we present  two analogues of \eqref{eq:stembridge}
using excedance statistics for permutations. 



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Background and preliminaries}\label{sec:2}

\subsection{Permutation statistics and two bijections}	
	
For  $\sigma=\sigma(1)\sigma(2)\cdots\sigma(n)\in \S_{n}$  with convention 0--0, i.e., $\sigma(0)=\sigma(n+1)=0$,  a value $\sigma(i)$ ($1\leq i\leq n$) is called
\begin{itemize}
\item a \emph{peak} if $\sigma(i-1)&lt;\sigma(i)$ and $\sigma(i)&gt;\sigma(i+1)$;
\item a \emph{valley} if $\sigma(i-1)&gt;\sigma(i)$ and $\sigma(i)&lt;\sigma(i+1)$;
\item a \emph{double ascent} if $\sigma(i-1)&lt;\sigma(i)$ and $\sigma(i)&lt;\sigma(i+1)$;
\item a \emph{double descent} if $\sigma(i-1)&gt;\sigma(i)$ and $\sigma(i)&gt;\sigma(i+1)$.
\end{itemize}


The set of Peaks (resp. Valleys, double ascents, double descents) of $\sigma$
is denoted by 
$$\Peak\sigma \quad  (\text{resp.} \; \Val\sigma,\; \Da\sigma,\;\Dd\sigma). $$
Let $\peak\,\sigma$ (resp. $\valley\,\sigma$, $\da\,\sigma$, $\dd\,\sigma$) 
be the number of peaks (resp. valleys, double ascents, double descents) of $\sigma$. 
For  $i\in [n]:=\{1, \ldots, n\}$, we introduce the following 
statistics:
\begin{equation}\label{31-2}
\begin{array}{ll}
(\cab)_i\:\sigma&amp;=\#\{j:  1&lt;j&lt;i \text{ and } \sigma(j)&lt;\sigma(i)&lt;\sigma(j-1)\}\\
(\bca)_i\:\sigma&amp;=\#\{j:  i&lt;j&lt;n \text{ and } \sigma(j+1)&lt;\sigma(i)&lt;\sigma(j)\}\\
(\bac)_i\:\sigma&amp;=\#\{j: i&lt;j&lt;n \text{ and } \sigma(j)&lt;\sigma(i)&lt;\sigma(j+1) \}\\
(\acb)_i\:\sigma&amp;=\#\{j: 1&lt;j&lt;i \text{ and } \sigma(j-1)&lt;\sigma(i)&lt;\sigma(j) \}
\end{array}
\end{equation}
and define (see \eqref{def:vincular}):
\[
(\cab)=\sum_{i=1}^n(\cab)_i, \quad
(\bca)=\sum_{i=1}^n(\bca)_i, \quad
(\bac)=\sum_{i=1}^n(\bac)_i, \quad
(\acb)=\sum_{i=1}^n(\acb)_i.
\]


Now, we consider $\sigma\in \S_{n}$ as a bijection 
$i\mapsto \sigma(i)$ for $i\in [n]$, a  value $x=\sigma(i)$  is called
 a \emph{cyclic peak}  if $i=\sigma^{-1}(x)&lt;x$ and $x &gt;\sigma(x)$;
a \emph{cyclic valley}  if $i=\sigma^{-1}(x)&gt;x$ and $x&lt;\sigma(x)$;
a \emph{double excedance} if $i=\sigma^{-1}(x)&lt;x$ and $x&lt;\sigma(x)$; 
 a  \emph{double drop} if $i=\sigma^{-1}(x)&gt;x$ and $x&gt;\sigma(x)$;
 a \emph{fixed point}  if $x=\sigma(x)$.
We say that $i\in[n-1]$ is an \textit{ascent}  of $\sigma$
 if $\sigma(i)&lt;\sigma(i+1)$ 
 and that $i\in[n]$ is a \textit{drop} 
  of $\sigma$
if $\sigma(i)&lt;i$.
Let 
$\Cpeak$ (resp. $\Cval$, $\Cda$,  $\Cdd$,  $\Fix$, $\Asc$, $\Drop$) be  the set  of $\emph{cyclic peaks}$
(resp.  $\emph{cyclic valleys}$, $\emph{ double excedances}$, $\emph{double drops}$, $\emph{fixed points}$, $\emph{ascents}$, $\emph{drops}$) and 
denote the corresponding cardinality by  
$\cpeak$ (resp. $\cval$, $\cda$, $\cdd$, $\fix$, $\asc$, $\drop$).
Moreover,
define 
\begin{align}
\cros_i\:\sigma &amp;=\#\{j:\:j&lt;i&lt;\sigma(j)&lt;\sigma(i)\quad\textrm{or}\quad\sigma(i)&lt;\sigma(j)\leq i&lt;j\},\label{def:cros}\\
\nest_i\:\sigma &amp;=\#\{j:\:j&lt;i&lt;\sigma(i)&lt;\sigma(j)\quad\textrm{or}
\quad\sigma(j)&lt;\sigma(i)\leq i&lt;j\}.\label{def:nest}
\end{align} 

Let 
$\cros\footnotemark =\sum_{i=1}^n\cros_i$ and $\icr\:\sigma =\cros\:\sigma^{-1}$.
\footnotetext{Our definition of $\cros$ 
corresponds to $\icr$ in \cite{FTHZ19}.}
Define  $\nest=\sum_{i=1}^n\nest_i$. Note (cf. \cite[Remark 2.4]{FTHZ19})
that 
 \begin{align}\label{nest=nest-1}
 \nest\:\sigma^{-1}=\nest\:\sigma\quad \text{for}\quad  \sigma\in \S_n.
 \end{align}
A pair of integers $(i,j)$ is an \emph{inversion} of $\sigma\in \S_n$ if 
$i&lt;j$  and $\sigma(i)&gt;\sigma(j)$, and    $\sigma(i)$ (resp. $\sigma(j)$) 
is called  \emph{inversion top} (resp. \emph{bottom}).
Let $ \inv\:\sigma $ be 
the inverion number of $\sigma$.

 For $\sigma\in \S_n$ with convention 0--$\infty$, i.e., 
  $\sigma(0)=0$ and $\sigma(n+1)=\infty$,
          let 
           $\Lpeak$ (resp. $\Lval$, $\Lda$, $\Ldd$) be  the set of $\emph{peaks}$
           (resp.  $\emph{valleys}$,         $\emph{double ascents}$ and $\emph{double decents}$) and 
           denote the corresponding cardinality by  $\lpeak$
           (resp. $\lvalley$, $\lda$ and $\ldd$).
          For $i\in [n]$, the value $\sigma(i)$ is called a {\em left-to-right maximum} 
if $\sigma(i)=\textrm{max}\:\{\sigma(1),\sigma(2),\ldots,\sigma(i)\}$           
          A double ascent $\sigma(i)$ $(i=1, \ldots, n)$ is called a \emph{foremaximum} 
 of $\sigma$ if it is at the same time a left-to-right maximum 
Denote the number of foremaxima 
of $\sigma$ by $\fmax\:\sigma$. 
  Note that  for the peak number $\peak'$ in \eqref{eq:peak-poly} we have  
 following identities~:
    \begin{align}\label{peak-val}
    \peak'=\val=\peak-1 \quad \text{and}\quad  \lvalley=\lpeak.
    \end{align}  
     

  
        Now we recall two bijections $\Phi$ and $\Psi$ due to
         Clarke et al.~\cite{CSZ97} and Shin--Zeng~\cite{SZ12}, respectively.        
   \subsection{The bijection $\Phi$} \label{def-Phi}    
        Let  $\sigma=\sigma(1)\ldots \sigma(n)\in \S_n$,
        an {\em inversion top number} (resp. {\em inversion bottom number}) of a letter $x:=\sigma(i)$
        in the word $\sigma$ is the number of occurrences of inversions of form $(i,j)$ (resp $(j,i)$).
        A letter  $\sigma(i)$ is a {\em descent top} (resp. {\em descent bottom}) if $\sigma(i)&gt;\sigma(i+1)$ (resp. $\sigma(i-1)&gt;\sigma(i)$).
          Given a permutation $\sigma$, we first construct two biwords, $f \choose f'$ and $g \choose g'$,  where 
        $f$ (resp. $g$) is the subword of descent bottoms (resp. nondescent bottoms) in $\sigma$ ordered increasingly, 
        and $f'$ (resp. $g'$)
        is the permutation of descent tops (resp. nondescent tops)  in $\sigma$ such that the inversion bottom (resp. top) number of each 
         letter $x:=\sigma(i)$ in $f'$ (resp. $g'$) is  $(\bca)_x\sigma$, and then form the biword $w=\left({f \atop f'}~{g \atop g'}\right)$ by concatenating $f$ and $g$, and $f'$ and $g'$, respectively.

Rearranging the columns of $w$, so that the bottom row is in increasing order, we obtain the permutation $\tau=\Phi(\sigma)$ as the top row of the rearranged bi-word.
      
  \medskip            
The following result can be found in \cite[Theorem 12]{SZ12} and its proof.
         
       \begin{lem}[Shin--Zeng]\label{97csz}
              For $\sigma\in \S_{n}$, we have
       	\begin{align}\label{eq:key}
       	&amp;(\bca, \cab, \des,\asc, \lda-\fmax, \ldd, \lvalley, \lpeak,\fmax)\sigma\nonumber\\
	=&amp;
        ( \nest, \icr, \drop, \exc+\fix, \cda, \cdd, \cvalley,\cpeak,\fix)\Phi(\sigma)\\
     =&amp;
       ( \nest, \cros, \exc, \drop+\fix, \cdd, \cda, \cvalley,\cpeak,\fix)(\Phi(\sigma))^{-1},\nonumber
       	\end{align}
       	\begin{align}\label{eq:keycsz}
       	(\Lval, \Lpeak, \Lda, \Ldd)\sigma=
       	( \Cval, \Cpeak, \Cda\cup\Fix, \Cdd)\Phi(\sigma),
       	\end{align}	
       	\begin{equation}\label{231nest}
       	(\bca)_i\sigma = \nest_i\Phi(\sigma) \quad \forall i=1,\ldots,n.
       	\end{equation}
              \end{lem}

   \subsection{The bijection  $\Psi$}   \label{def-Psi}
        Given a permutation $\sigma \in \S_n$, let 
 \begin{equation} \label{eq:hat}
\hat{\sigma}=\begin{pmatrix}
1&amp; 2&amp;\ldots&amp; n&amp;n+1\\
\sigma(1)+1&amp;\sigma(2)+1&amp;\dots&amp;\sigma(n)+1&amp;1
\end{pmatrix},
\end{equation}
         and $\tau:=\Phi(\hat{\sigma}) \in \S_{n+1}$.
        Since the last element of $\hat{\sigma}$ is $1$, 
 the first element  of $\tau$ should be $n+1$. Define the bijection $\Psi: \S_n\to \S_n$ by 
        \begin{align}\label{def:Psi}
        \Psi(\sigma):=\tau(2)\dots\tau(n+1) \in \S_{n}.
        \end{align}
        
       
        
\subsection{The star variation}
For $\sigma=\sigma(1)\cdots\sigma(n) \in \S_n$,  we define
its \emph{star compagnon}
$\sigma^*$ as a permutation of $\{0, \ldots, n\}$ by
\begin{equation}
\label{eq:star}
\sigma^*=\begin{pmatrix}
0&amp; 1&amp;2&amp;\ldots&amp; n\\
n&amp; \sigma(1)-1&amp;\sigma(2)-1&amp;\dots&amp;\sigma(n)-1
\end{pmatrix}.
\end{equation}          
       We define the following sets of cyclic star statistics for $\sigma$:        \begin{align}
       \Cpeak^*\sigma &amp;=   \{i \in [n-1]:  (\sigma^{*})^{-1}(i)&lt;i&gt;\sigma^{*}(i)\},\label{Eq.3.3}\\
       \Cval^{*}\sigma &amp;=   \{i \in [n-1]:  (\sigma^{*})^{-1}(i)&gt;i&lt;\sigma^{*}(i)\},\\
       \Cda^{*}\sigma &amp;=   \{i \in [n-1]: (\sigma^{*})^{-1}(i)&lt;i&lt;\sigma^{*}(i)\},\label{Eq.3.8}\\
       \Cdd^{*}\sigma &amp;=   \{i \in [n-1]: (\sigma^{*})^{-1}(i)&gt;i&gt;\sigma^{*}(i)\},\label{Eq.3.7}\\
       \Fix^{*}\sigma &amp;=  \{i\in [n-1]: i=\sigma^{*}(i)\},\label{Eq.3.4}\\
       \Wex^{*}\sigma &amp;= \{i\in [n-1]: i\le\sigma^{*}(i)\} (= \exc\, \sigma),\\
       \Drop^{*}\sigma &amp;=  \{i\in [n]: i&gt;\sigma^{*}(i)\}.\label{Eq.3.9}
\end{align}
The corresponding cardinalties are denoted by 
$\cpeak^*$, $\cval^*$, $\cda^*$,  $\cdd^*$,  $\fix^*$, $\wex^*$ and 
$\drop^*$, respectively.

By \eqref{Eq.3.3}, \eqref{Eq.3.7} and \eqref{Eq.3.9},
 we have $\drop^*-1=\cdd^*+\cpeak^*.$

\begin{thm}\label{SZsetprop}
          For $\sigma\in\S_n$, we have 
          	\begin{align}\label{set-stat}
          	(\Val,\Peak\setminus \{n\},\Da,\Dd)\sigma=
	(\Cval^*,\Cpeak^*,\Cda^*\cup\Fix^*,\Cdd^*)\Psi(\sigma)
          \end{align}
   and 
    \begin{align}\label{nest-cros-i}
  ((\bac)_i,\;(\cab)_i)\sigma=  (\nest_i, \; \cros_i)\Psi(\sigma)
          		\quad\text{for}\quad  i\in [n].
		\end{align}
          \end{thm}
        
      

    Since $\asc=\val+\da$, $\des=\peak+\dd-1$, $\wex^*=\cval^*+\cda^*+\fix^*$, $\drop^*-1=\cdd^*+\cpk^*$,
    we get the following result in \cite[Theorem 12]{SZ12}. 
    \begin{cor}[Shin--Zeng]\label{SZbijection} For $\sigma\in \S_n$ we have 
    	\begin{align}
    	&amp;(\bac, \cab, \des, \asc, \da, \dd, \valley)\sigma\nonumber\\
    	&amp;=(\nest, \cros,\drop^*-1,\wex^*,\cda^*+\fix^*,\cdd^*,\cvalley^*)\Psi(\sigma).\label{starCSZ97}
    	\end{align}
    \end{cor}
    
    
        \subsection{Laguerre histories as permutation encodings}
		A 2-\emph{Motzkin path}  of length $n$  is a word $\mathbf{s}:=s_1\ldots s_n$ on the alphabet $\{\su, \sd, \la, \lb\}$ 
		such that  $|s_1\ldots s_n|_\su= [s_1\ldots s_i|_\sd$ and 
		\begin{equation}\label{heigt-condition}
		h_{i}:=|s_1\ldots s_i|_\su-[s_1\ldots s_i|_\sd\geq 0 \quad (i=1, \ldots, n),
		\end{equation}
	 where $|s_1\ldots s_i|_\su$ is the number of letters $\su$ in the word $s_1\ldots s_i$. 

		A \emph{Laguerre history} (resp. \emph{restricted Laguerre history}) of length  $n$ is  a pair $(\mathbf{s, p})$, where $\mathbf{s}$ is 
		a 2-Motzkin path $s_1\ldots s_n$ 
		and $\mathbf{p}=(p_1, \ldots, p_n)$ with $0\leq p_i\leq h_{i-1}$
		 (resp. $0\leq p_i\leq h_{i-1}-1$ if $s_i=\lb$ or D) and $h_0=0$.
		Let $\LH_n$ (resp. $\LH^*_n$) be the set of Laguerre histories 
		(resp. restricted Laguerre histories)
		of length $n$.
		There are several  well-known  bijections 
		between $\S_{n}$ and $\LH^*_{n}$ and 
		$\LH_{n-1}$, see \cite{DV94, CSZ97} and references therein.
 
    
			\subsection{Fran\c con--Viennot bijection}
		We recall a version of Fran\c con and Viennot's bijection 
		$\psi_{{FV}}: \S_{n+1}\to \LH_{n}$. Given $\sigma\in \S_{n+1}$,  the Laguerre history $(\mathbf{s, p})$ is defined as follows: 
		\begin{align}\label{FV}
		s_i=U\; (\textrm{resp.} D, \; \la,\; \lb)\;  \textrm{if} \; i\in \Val\sigma \; 
		(\textrm{resp.} \; i\in \Peak\sigma, \;
		i\in \Da\sigma, \; i\in \Dd\sigma)
		\end{align}
		and  $p_i=(\bac)_i~\sigma$ for $i=1, \ldots, n$.
				
	\medskip	

    \begin{thm}\label{HMZ}
    The mapping $\psi:=\psi_{{FV}}\circ \Psi^{-1}$ is a bijection
    from  $\S_{n+1}$ to  $\LH_n$. If $\psi(\sigma)=(\mathbf{s, p})$ with  $\sigma\in \S_{n+1}$,
    then,  for $i=1, \ldots, n$,
    \begin{align}\label{2.35}
    s_i=U\; (\textrm{resp.} D, \; \la,\; \lb)\;  \textrm{if} \; i\in \Cval^*\sigma \; 
		(\textrm{resp.} \; i\in \Cpeak^*\sigma, \;
		i\in \Cda^*\sigma\cup\Fix^*\sigma, \; i\in \Cdd^*\sigma)    
     \end{align}
    with $p_i=\nest_i\sigma$. 
    
   \end{thm} 
   \begin{proof} This follows from \cref{SZsetprop} by comparing 
   \eqref{2.35} with  \eqref{FV}, see  the commutative diagram in \cref{fig:1}.
    \end{proof}
  
   \begin{cor}\label{starpqzhuang}
The two sextuple  statistics 
   $$
   (\nest, \cros, \exc, \cdd^*,\cda^*+\fix^*, \cpeak^*)\quad \textrm{and}\quad
   (\bac, \cab, \des,  \da, \dd, \peak-1)
   $$ 
    		are equidistributed on $\S_n$.
    	\end{cor}

        %%%%%%%%%%%%%%%%%%%
             We recall two bijections $\phi_{{FZ}}$ and $\phi_{{FV}}$ from $\S_{n}$ to $\L_n^*$.  
             
 \subsection{Restricted Fran\c con--Viennot  bijection} 
 	We recall a restricted version of Fran\c con and Viennot's bijection 
		$\phi_{{FV}}: \S_{n}\to \LH_{n}^*$. Given $\sigma\in \S_{n}$,  the Laguerre history $(\mathbf{s, p})$ is defined as follows: 
		\begin{align}\label{res-FV}
		s_i=U\; (\textrm{resp.} D, \; \la,\; \lb)\;  \textrm{if} \; i\in \Lval\sigma \; 
		(\textrm{resp.} \; i\in \Lpeak\sigma, \;
		i\in \Lda\sigma, \; i\in \Ldd\sigma)		
		\end{align}
		and  $p_i=(\bca)_i~\sigma$ for $i=1, \ldots, n$.
 

       \subsection{Foata--Zeilberger  bijection}
      %%%%%%%%%%%%%%%%%%%%
       This  bijection  $\phi_{{FZ}}$ encodes permutations 
       using  cyclic statistics.
       Given $\sigma\in \S_{n}$, 
       $\phi_{{FZ}}: \S_{n}\to \L_n^*$ is 
       for $i=1, \ldots, n$,
       \begin{align}\label{resFZ}
       	s_i=U\; (\textrm{resp.} D, \; \la,\; \lb)\;  \textrm{if} \; i\in \Cval\sigma \; 
		(\textrm{resp.} \; i\in \Cpeak\sigma, \;
		i\in \Cda\sigma\cup\Fix\sigma, \; i\in \Cdd\sigma)		       
        \end{align}
       with $p_i=\nest_i\sigma$.
              By \eqref{eq:keycsz} and \eqref{231nest}, we can build a comutative diagram,
       see the right diagram of Figure 1. 
       
       
   By contracting the  continued fraction~\eqref{gf:eulerian}  
we derive   the two J-type continued fraction formulae (cf. \cite{Fla80})
\begin{align}\label{lag1}
\sum_{n\geq 0} A_{n+1}(t) z^n
=\cfrac{1}{1-(1+t)\cdot z-\cfrac{1\cdot2\cdot t\cdot z^{2}}{1-2(1+t)\cdot z-\cfrac{2\cdot3\cdot t\cdot z^{2}}{1-\cdots}}}
\end{align}
and 
\begin{align}\label{lag2}
\sum_{n\geq 0} A_n(t) z^n=\cfrac{1}{1-(1+0\cdot t)\cdot z-\cfrac{1^2\cdot z^{2}}{1-(2+1\cdot t)\cdot z-\cfrac{2^2\cdot t\cdot  z^{2}}{1-\cdots}}}.
\end{align}

         %%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}\label{FVBFZB}
\begin{center}

\begin{tikzpicture}[scale=0.3]

\node (LH) at (-12,10) {$\LH^*_n$};

\node (SL) at (-20,0) {$\S_{n}$};

\node (SC) at (-5,0) {$\S_{n}$};

\node (1) at (-18, 5){$\phi_{{FV}}$};

\node (2) at (-6, 5){$\phi_{{FZ}}$};

\node (3) at (-12, 1) {$\Phi$}; 
\begin{scope}[font=\footnotesize,-&gt;]

\draw (SL) --  (LH);

\draw (SC) -- (LH);

\draw (SL) -- (SC);

\end{scope}
\end{tikzpicture}
\hspace{2cm}
\begin{tikzpicture}[scale=0.3]

\node (LH) at (-12,10) {$\LH_n$};

\node (SL) at (-20,0) {$\S_{n+1}$};

\node (SC) at (-5,0) {$\S_{n+1}$};

\node (1) at (-18, 5){$\psi_{{FV}}$};

\node (2) at (-6, 5){$\psi$};

\node (3) at (-12, 1) {$\Psi$}; 
\begin{scope}[font=\footnotesize,-&gt;]

\draw (SL) -- (LH);

\draw (SC) -- (LH);

\draw (SL) --(SC);
\end{scope}

\end{tikzpicture}

\end{center}
\caption{
Bijections  $\Phi=\phi_{FZ}^{-1}\circ\phi_{FV}$ and $\psi= \psi_{FV}\circ \Psi^{-1}$.  
}\label{fig:1}
\end{figure}
 
In view of  Flajolet's combinatorial interpretation in terms of 
weighted Motzkin paths
 for generic J-type continued fraction expansions~\cite{Fla80}, 
 Fran\c con--Viennot's bijection $\psi_{FV}$ (resp. its restricted version $\phi_{FV}$) between permutations and \emph{Laguerre histories} provides  a bijective proof of \eqref{lag1} (resp. \eqref{lag2}), while 
Foata--Zeilberger's bijection $\psi_{FZ}$~\cite{FZ90} 
gives a bijective proof of  \eqref{lag2}. 
More precisely,
Fran\c con--Viennot~\cite{FV79}  set up 
a bijection (and its restricted version) 
from permutations to \emph{Laguarre histories} using  
\emph{linear statistics of permutation}, while 
Foata--Zeilberger's bijection~\cite{FZ90}  uses 
\emph{cyclic statistics of permutations}. 
Clarke-Steingr\'imsson-Zeng~\cite{CSZ97} gave a direct bijection  $\Phi$
on permutations  converting  statistic $\des$ into $\exc$  on permutations, 
 and linking  the restricted  Fran\c con--Viennot's bijection $\phi_{FV}$ to  Foata--Zeilberger bijection $\phi_{FZ}$, see \cref{fig:1}.  As a variation of  $\Phi$, Shin and Zeng~\cite{SZ12} constructed  a bijection $\Psi$ on permutations to derive a
 cycle version of linear statistics on permutations, which are obtained 
via  Fran\c con--Viennot bijection $\psi_{FV}$. 
One of our main results (cf. \cref{HMZ}) shows  that a  direct  description of  the bijection
 $\psi:= \psi_{FV}\circ \Psi^{-1}$ from $\S_{n+1}$ to $\LH_n$
 is straightforward.

		
		
		

%%%%%
%%%%%




%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Pattern avoidances and 2-Motzkin paths}

A permutation  $\sigma$ is called 231-\emph{avoiding permutation} if 
there is no triple of indices $i&lt;j&lt;k$ such that $\sigma(k)&lt;\sigma(i)&lt;\sigma(j)$. The Narayana polynomials are defined by
$$
N_n(t)=\sum_{\sigma\in \S_n(231)}t^{\des\,\sigma},
$$
where $\S_n(231)$ is the set of 231-avoiding permutations in $\S_n$.
It is well known that Narayana polynomial is $\gamma$-positive and 
 have the  expansion~\cite[Chapter 4]{Pet15}:
\begin{align}\label{nara-gamma}
N_n(t)=\sum_{k=0}^{n/2}\widetilde{\gamma}_{n,j}t^j (1+t)^{n-1-2j},
\end{align}
where $\widetilde{\gamma}_{n,j}=|\{\sigma\in \S_n(231): \;\des(\sigma)=\peak(\sigma)=j\}|$.
%$$%%%%%%%%%%%%%%%
As for Eulerian polynomials, by contraction, from 
\begin{align}
\sum_{n\geq 0} N_n(t) z^n= 1/1-z/1-t\cdot z/1-z/1-t\cdot z/1-\ldots
\end{align}
we derive immediately  the following continued fractions 
\begin{align}\label{2-motzkin}
\sum_{n\geq 0} N_{n+1}(t) z^n=\cfrac{1}{1-(1+t)\cdot z-\cfrac{t\cdot z^{2}}{1-(1+t)\cdot z-\cfrac{
 t\cdot z^{2}}{1-\cdots}}}
\end{align}
and 
\begin{align}\label{2-motzkin-star}
\sum_{n\geq 0} N_n(t) z^n=\cfrac{1}{1-z-\cfrac{ t\cdot z^{2}}{1-(1+t)\cdot z-\cfrac{t\cdot z^{2}}{1-(1+t)\cdot z-\cdots}}}.
\end{align}
Note that $N_n(1)$ is the n-th Catalan number $C_n=\frac{1}{n+1}{2n\choose n}$, there are several well-known $q$-Narayana polynomials in the literature, see \cite{CEKS13} and \cite{FTHZ19}.



Given two permutations $\sigma\in\S_n$ and $\tau\in\S_k$, we say that \emph{$\sigma$ contains the pattern $\tau$} if there exists a set of indices $1\le i_1&lt;i_2&lt;\cdots&lt;i_k\le n$ such that the subsequence $\sigma(i_1)\sigma(i_2)\cdots\sigma(i_k)$ of $\sigma$ is order-isomorphic to $\tau$. Otherwise, $\sigma$ is said to \emph{avoid $\tau$}. For example, the permutation $15324$ contains the pattern  $321$ and avoids  the pattern
$231$. The set of permutations of length $n$ that avoid patterns $\tau_1,\tau_2,\cdots,\tau_m$ is denoted as $\S_n(\tau_1,\tau_2,\cdots,\tau_m)$.

Moreover  we shall consider the so-called \emph{vincular patterns}. The number of occurrences of vincular patterns $\cab$, $\bca$, $\bac$ and $\acb$ in $\pi\in \S_n$ are defined (cf. \eqref{31-2}) by
\begin{equation}\label{def:vincular}
\begin{array}{ll}
(\cab)\:\pi&amp;=\#\{(i,j):  i+1&lt;j\le n \text{ and } \pi(i+1)&lt;\pi(j)&lt;\pi(i)\},\\
(\bca)\:\pi&amp;=\#\{(i,j):  j&lt;i&lt;n \text{ and } \pi(i+1)&lt;\pi(j)&lt;\pi(i)\},\\
(\bac)\:\pi&amp;=\#\{(i,j): j&lt;i&lt;n \text{ and } \pi(i)&lt;\pi(j)&lt;\pi(i+1) \},\\
(\acb)\:\pi&amp;=\#\{(i,j): i+1&lt;j\le n \text{ and } \pi(i)&lt;\pi(j)&lt;\pi(i+1) \}.
\end{array}
\end{equation}
Similarly, we use $\S_n(\cab)$ to denote the set of permutations of length $n$ that avoid the vincular pattern $\cab$, etc. 
In order to apply  Laguerre history 
to  count pattern-avoiding permutations,  we will need the following results in
\cite[Lemmas 2.8 and 2.9]{FTHZ19}.
\begin{lem}[{\cite[Lemma 2.8]{FTHZ19}}]\label{general-231-321}
For any $n\geq1$, we have
\begin{align}
\S_n(\bac)&amp;=\S_n(213), \quad
\S_n(\cab)=\S_n(312),\label{eq1} \\
\S_n(\acb)&amp;=\S_n(132), \quad
\S_n(\bca)=\S_n(231). \label{eq4}
\end{align}
\end{lem}

\begin{lem}[{\cite[Lemma~2.9]{FTHZ19}}]\label{321:nest}
\begin{itemize}
\item[(i)] A permutation $\pi\in \S_n$ belongs to $\S_n(321)$ if and only if $\nest\pi=0$ 
\item[(ii)] The mapping $\varPhi$ has the property that  $\varPhi(\S_{n}(231))=\S_n(321)$. 
\end{itemize}
\end{lem}

We use  $\mathcal{CM}_n$ to denote  the set of 2-Motzkin paths
of length $n$ and $\mathcal{CM}^{*}_n$ to denote its subset that is composed 
of 2-Motzkin paths without $\lb$-step at level zero, i.e., 
if $h_{i-1}=0$, then $s_i\neq \lb$. Noticing that the generating function $\sum_{n\geq 0}|\mathcal{CM}^{*}_n|z^n$ has the continued fraction expansion \eqref{2-motzkin-star} with $t=1$,  we derive that $|\mathcal{CM}^{*}_n|=C_n$. Similarly, by \eqref{2-motzkin} we see that  $|\mathcal{CM}_n|=C_{n+1}$.



Let $\widetilde{\phi}_{FV}$ (resp. $\widetilde{\phi}_{FZ}$, $\widetilde{\psi}_{FV}$,  
$\widetilde{\psi}$, $\widetilde{\Psi}$, $\widetilde{\Phi}$) be the restriction of $\phi_{FV}$ (resp. $\phi_{FZ}$, $\psi_{FV}$, 
$\psi$, $\Psi$, $\Phi$) on the sets  $\S_n(231)$ (resp. $\S_n(321)$, $\S_{n+1}(213)$, $\S_{n+1}(321)$, $\S_{n+1}(213)$, $\S_n(231)$).
By \cref{general-231-321,321:nest} and \cref{fig:1}, we obtain the diagrams in \cref{fig:2}.


\begin{figure}\label{pattern-digram}
\begin{center}

\begin{tikzpicture}[scale=0.3]

\node (LH) at (-12,10) {$\mathcal{CM}_n$};

\node (SL) at (-20,0) {$\S_{n+1}(213)$};

\node (SC) at (-5,0) {$\S_{n+1}(321)$};

\node (1) at (-18, 5){$\widetilde{\psi}_{{FV}}$};

\node (2) at (-6, 5){$\widetilde{\psi}$};

\node (3) at (-12, 1) {$\widetilde{\Psi}$}; 
\begin{scope}[font=\footnotesize,-&gt;]

\draw (SL) -- (LH);

\draw (SC) -- (LH);

\draw (SL) --(SC);

\end{scope}
\end{tikzpicture}
\hspace{1cm}%%
\begin{tikzpicture}[scale=0.3]
\node (LH) at (-12,10) {$\mathcal{CM}^{^*}_n$};
\node (SL) at (-20,0) {$\S_{n}(231)$};

\node (SC) at (-5,0) {$\S_{n}(321)$};

\node (1) at (-18, 5){$\widetilde{\phi}_{{FV}}$};

\node (2) at (-6, 5){$\widetilde{\phi}_{{FZ}}$};

\node (3) at (-12, 1) {$\widetilde{\Phi}$}; 
\begin{scope}[font=\footnotesize,-&gt;]

\draw (SL) --  (LH);

\draw (SC) -- (LH);

\draw (SL) -- (SC);

\end{scope}
\end{tikzpicture}
\end{center}
\caption{Bijections $\widetilde{\Psi}=\widetilde{\psi}^{-1}\circ\widetilde{\psi}_{FV}$  
and $\widetilde{\Phi}=\widetilde{\phi}_{FZ}^{-1}\circ\widetilde{\phi}_{FV}$.}
\label{fig:2}
\end{figure}


\section{Main results}\label{sec:3}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

For  a finite set of permutations  $\Omega$ and  
$m$ statistics  $\stat_1,  \ldots, \stat_m$  on $\Omega$, we define the generating polynomial 
\begin{align}
P^{(\stat_1,  \ldots,  \stat_m)}(\Omega; t_1, \ldots,  t_m):=&amp;\sum_{\sigma\in\Omega}
t_1^{\stat_1\,\sigma}\,\ldots\, t_m^{\stat_m\,\sigma}.
\end{align}

\begin{thm} \label{thm:cpeakS_1}
For $n\geq 1$,  
\begin{align}\label{cpeakS_1}
&amp;P^{(\nest, \cros,  \exc, \fix)}(\S_n; p, q, tq, r)\nonumber\\
=&amp;\left(\frac{1+xt}{1+x}\right)^{n}P^{(\nest, \cros,\cpeak, \exc, \fix)}
\left(\S_n; p, q, \frac{(1+x)^{2}t}{(x+t)(1+xt)},
\frac{q(x+t)}{1+xt}, \frac{(1+x)r}{1+xt}\right),
\end{align}
equivalently, 
\begin{gather}
P^{(\nest, \cros, \cpeak, \exc, \fix)}(\S_n; p, q, x, qt, r)\nonumber\\
=\left(\frac{1+u}{1+uv}\right)^{n}P^{(\nest, \cros,  \exc, \fix)}\left(\S_n; p, q, qv, \frac{(1+uv)r}{1+u}\right),\label{cpeakS_2}
\end{gather}
where $u=\frac{1+t^{2}-2xt-(1-t)\sqrt{(1+t)^{2}-4xt}}{2(1-x)t}$ and
$v=\frac{(1+t)^{2}-2xt-(1+t)\sqrt{(1+t)^{2}-4xt}}{2xt}.$
\end{thm}

\begin{remark}
   Cooper et al.~\cite[Theorem~11]{CJZ19} have recently proved
  the $p=q=1$ case of \eqref{cpeakS_1} by applying  Sun and 
Wang's CMFS action~\cite{SW14}.
\end{remark}

We define the polynomial
\begin{align}\label{C1eq:dfA1-spec}
A_n(p,q,t) := 
\sum_{\sigma\in \S_n} p^{\nest\, \sigma} q^{\cros \,\sigma}  t^{\exc\,\sigma}.
\end{align}
The following is a 
generalization of Stembridge's identity~\eqref{eq:stembridge}.
 \begin{thm}\label{pqpk}
 For $n\geq1$, we have 
\begin{equation}\label{eqpk1}
A_{n}(p, q, t)=\left(\frac{1+xt}{1+x}\right)^{n-1}P^{(\nest, \cros,\cpk^*,\exc)}
\left(\S_n; p, q, \frac{(1+x)^{2}t}{(x+t)(1+xt)},\frac{x+t}{1+xt}\right),
\end{equation}
equivalently, 
\begin{equation}
P^{(\nest, \cros, \cpk^*,\exc)}(\S_n;  p, q, x,t)=\left(\frac{1+u}{1+uv}\right)^{n-1}A_{n}(p, q, v),\label{eupk3}
\end{equation}
where $u=\frac{1+t^{2}-2xt-(1-t)\sqrt{(1+t)^{2}-4xt}}{2(1-x)t}$ and
$v=\frac{(1+t)^{2}-2xt-(1+t)\sqrt{(1+t)^{2}-4xt}}{2xt}.$
\end{thm}

%
\begin{remark}  
By \cref{starpqzhuang} and \eqref{eqpk1},  when $x=1$ or $p=q=1$
we recover two special cases of \eqref{eqpk1} due to 
 Br\"{a}nd\'{e}n's result \cite[(5.1)]{Bra08} and  Zhuang~\cite[Theorem $4.2$]{Zhuang17}, respectively.
\end{remark}


\acknowledgements{
Part of this work was done during the first and second authors' studies at Institut Camille Jordan,  Universit\'e Claude Bernard Lyon 1.
}
\printbibliography



\end{document}
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