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\newtheorem{theorem}{Theorem}[section]
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%\usepackage{lipsum}
%\usepackage{color}
\newcommand\mycom[2]{\genfrac{}{}{0pt}{}{#1}{#2}}

\def\blue{\textcolor{blue}}
\def\red{\textcolor{red}}


\def\S{\mathfrak{S}}
\def\PRW{\mathcal{M}}
\def\N{\mathbb{N}}
\def\A{\mathcal{A}}
\def\Z{\mathcal{Z}}
\def\Web{\mathcal{CA}}
\def\s{\sigma}
\def\a{\alpha}
\def\be{\beta}
\def\uu{\textbf{u}}
\DeclareMathOperator\cyc{cyc}
\DeclareMathOperator\exc{exc}
\DeclareMathOperator\drop{drop}
\DeclareMathOperator\fix{fix}
\DeclareMathOperator\cpk{cpk}
\DeclareMathOperator\cval{cval}
\DeclareMathOperator\cda{cda}
\DeclareMathOperator\cdd{cdd}
\DeclareMathOperator\asc{asc}
\DeclareMathOperator\des{des}
\DeclareMathOperator\val{val}
\DeclareMathOperator\pk{pk}
\DeclareMathOperator\lda{da}
\DeclareMathOperator\da{da}
\DeclareMathOperator\rdd{dd}
\DeclareMathOperator\dd{dd}
\DeclareMathOperator\lrmax{lmax}
\DeclareMathOperator\rlmax{rmax}
\DeclareMathOperator\rlmin{rmin}
\DeclareMathOperator\rlminda{rminda}
\DeclareMathOperator\fmax{lmaxpk}
\DeclareMathOperator\bmax{rmaxpk}
\DeclareMathOperator\lrmaxda{lmaxda}
\DeclareMathOperator\rlmaxdd{rmaxdd}
\DeclareMathOperator\Cval{Cval}
\DeclareMathOperator\Cpk{Cpk}
\DeclareMathOperator\Cda{Cda}
\DeclareMathOperator\Cdd{Cdd}
\DeclareMathOperator\Fix{Fix}


%% define your title in the usual way
\title[Variations of the \texorpdfstring{$(\alpha,t)$}{(alpha,t)}-Eulerian polynomials and gamma positivity]{Variations of the \texorpdfstring{$(\alpha,t)$}{(alpha,t)}-Eulerian polynomials and gamma-positivity}

\author[C. Xu and J. Zeng]{\texorpdfstring{Chao Xu~\thanks{\href{mailto:xu@math.univ-lyon1.fr}{xu@math.univ-lyon1.fr}} \and Jiang Zeng~\thanks{\href{mailto:zeng@math.univ-lyon1.fr}{zeng@math.univ-lyon1.fr}}}{Chao Xu~ and Jiang Zeng~}}

%% then use \addressmark to match authors to institutions here
\address{Universite Claude Bernard Lyon 1, CNRS, Centrale Lyon, INSA Lyon, \\Université Jean Monnet, ICJ UMR5208, 69622 Villeurbanne, France}

%% put the date of submission here
\received{\today}
\abstract{
We define a multivariable generalization  of the Eulerian polynomials  using linear and  descent based statistics of permutations and establish the connection with the 
$(\alpha,t)$-Eulerian polynomials  based on cyclic  and excedance based statistics of permutations.
 As applications of this connection,
 we obtain the exponential generating function for the multivariable  Eulerian polynomials and 
 $\gamma$-positive  formulas of  two variants of Eulerian polynomials.
    We also show that enumerating the cycle André permutations with respect to the number of drops, fixed points and cycles gives rise to the normalised $\gamma$-vectors of the $(\alpha,t)$-Eulerian polynomials. 
  Our result generalizes and unifies several recent results in the literature. 
 }
% \resume{Nous définissons une généralisation multivariée des polynômes eulériens à l'aide de statistiques linéaires basées sur la descente de permutations  et  établissons le lien avec les $(\alpha,t)$-polynômes eulériens  définis à l'aide des statistiques cycliques  de permutations et basées sur l'excédance de permutations.
% En application de cette connexion, nous obtenons  la fonction génératrice exponentielle des polynômes eulériens multivariables et les formules $\gamma$-positives de deux variantes de polynômes eulériens.
% Nous montrons aussi  que l'énumération de  permutations cycliques d'André  par rapport au nombre de anti-excédances, de points fixes et de cycles donne naissance aux $\gamma$-vecteurs normalisés des polynômes $(\alpha,t)$-eulériens.
% Notre résultat généralise et unifie plusieurs résultats récents de la littérature.
%  }
\keywords{$(\alpha,t)$-Eulerlian polynomials, left-to-right maxima, cycle André permutations, cyclic valley-hopping, gamma-positivity}

\usepackage[backend=bibtex, giveninits]{biblatex}
\addbibresource{sample.bib}
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\begin{document}
\maketitle
\section{Introduction}
The Eulerian polynomials 
 have a long 
and rich history, some of which is given in   \cite{Fo10, Pe15,FS70,MHYY25}.
  For any positive integer $n$, 
we denote  the symmetric group of    $[n]:=\{1, 2,\ldots, n\}$ by $\S_n$.
For  $\s\in \S_n$, 
 the integer  $i\in [n-1]$ is called a \emph{descent} (\textbf{des})  if $\s(i)>\s(i+1)$; an \emph{ascent} (\textbf{asc})  if $\s(i)<\s(i+1)$; an \emph{excedance} (\textbf{exc}) if $i<\s(i)$.
It is well-known that the Eulerian polynomials $A_n(x)$ have 
the following combinatorial interpretations:
\begin{align}
A_n(x):=\sum_{k=0}^{n-1}\left\langle \mycom{n}{k}\right\rangle x^k=\sum_{\s\in \S_n}x^{\asc(\s)}=\sum_{\s\in \S_n}x^{\des(\s)}=\sum_{\s\in \S_n}x^{\exc(\s)}.
\end{align}
Let $\PRW_n$ be the set of permutations $\s\in\S_n$ such that the first descent (if any) of $\s$ appears at $\sigma^{-1}(n)$. The \emph{binomial-Eulerian polynomials} were introduced  by Postnikov, Reiner, and Williams~\cite[Section~10.4]{PRW08} as 
 the 
$h$-polynomials of stellohedrons, and can also be defined as in the following
\begin{equation}
\widetilde{A}_n(x):=\sum_{\sigma\in \PRW_{n+1}} x^{\des(\sigma)}=1+x\sum_{m=1}^n\binom{n}{m}A_m(x).
\end{equation}
It is well-known~\cite{FS70,PZ21}  that the Eulerian polynomials $A_n(x)$ have the following 
$\gamma$-positive expansion 
 \begin{align}
 A_{n+1}(x)&=\sum_{j=0}^{\lfloor n/2\rfloor} \gamma_{n,j}\,x^j(1+x)^{n-2j}\label{gamm-r-n-j}\\
 &=\sum_{j=0}^{\lfloor n/2\rfloor}2^j d_{n,j}\,x^j(1+x)^{n-2j},\label{euler-gamma}
 \end{align}
 where $\gamma_{n,j}$ is the number of permutations  without double descents having $j$ descents in $\S_{n+1}$ and  $d_{n,j}$ is the number of Andr\'e permutations  with $j$ descents in $\S_{n+1}$. 
It is also known~\cite[Section~10.4]{PRW08} that  the polynomials $\widetilde{A}_n(x)$ have the following gamma positive formula
\begin{equation}\label{PRW-positivity}
    \widetilde{A}_{n}(x)=\sum_{j=0}^{\lfloor n/2\rfloor} \widetilde{\gamma}_{n,j}\,x^j(1+x)^{n-2j},
\end{equation}
where $\widetilde{\gamma}_{n,j}$ is the number of $\sigma\in\PRW_{n+1}$ such that $\sigma$ has  $j$ descents and no double descents.

The coefficients of the polynomials $\widetilde{A}_n(x)$  can also be nicely expressed as sums of products of Eulerian numbers and binomial coefficients.  %Let $A_n(x)=\sum_{k=0}^{n-1}\left\langle \mycom{n}{k}\right\rangle x^k$. 
For integers $a, b\geq 0$,  Chung, Graham, and Knuth~\cite{CGK10} found the following 
identity with three proofs:
\begin{equation}\label{cgk}
\sum_{k\geq 0} \binom{a+b}{k}
\left\langle \mycom{k}{a-1}\right\rangle=
\sum_{k\geq 0}\binom{a+b}{k}
\left\langle \mycom{k}{b-1}\right\rangle,
\end{equation}
where $\left\langle\mycom{0}{0}\right\rangle=1$.
Shareshian and Wachs~\cite{SW20} noticed that the above identity corresponds exactly 
to the palindromicity  of  the coefficients of $\widetilde{A}_n(x)$. Recently, Ji and Lin~\cite[Theorem~4.1]{JL23} found an $\alpha$-analogue of~\eqref{cgk}.

For 
$\s\in \S_n$,  
 an index  $i\in [n]$ is a \emph{drop} (\textbf{drop}) of $\s$ if $i>\s(i)$; a \emph{fixed point} (\textbf{fix}) of $\s$ if $i=\s(i)$.
 We shall also consider a permutation $\sigma\in \S_n$ as a word
$\sigma=\sigma_1\ldots \sigma_n$ with $\sigma_i:=\sigma(i)$ for
$i\in [n]$.
Say that a letter $\s_i$ is a \emph{left-to-right maximum} (\textbf{lrmax}) of $\s$
if $\s_i>\s_j$ for every $j<i$; a \emph{right-to-left maximum} (\textbf{rlmax})  of $\s$ if $\s_i>\s_j$ for every  $j>i$.  

In the middle of 1970's 
Carlitz--Scoville considered  several multivariate 
 Eulerian polynomials,  among which  are the so-called $(\a, \be)$-Eulerian polynomials~\cite{CS74} 
 \begin{subequations} 
\begin{equation}\label{Carlitz-Scoville-alpha-beta-polynomials}
A_n(x,y\,|\, \a, \be):=\sum_{\s\in \S_{n+1}}
x^{\asc(\s)}y^{\des(\s)}\a^{\lrmax(\s)-1}\be^{\rlmax(\s)-1},
\end{equation}
and the  following  ones~\cite{CS77}, that we refer to \textbf{$(\a,t)$-Eulerian polynomials},  
\begin{align}\label{CS:def1}
A^{\cyc}_n(x,y, t\,|\,\alpha):=\sum_{\s\in \S_n} x^{\exc(\s)} y^{\drop(\s)}t^{\fix(\s)}\alpha^{\cyc(\s)},
\end{align}
\end{subequations} 
where  $\cyc(\s)$ denotes 
the number of cycles of $\s$. As   $y^nA^{\cyc}_n(x/y,1, t/y\,|\,\alpha)$ is equal to 
$A^{\cyc}_n(x,y, t\,|\,\alpha)$,
polynomial $A^{\cyc}_n(x,y, t\,|\,\alpha)$ is the homogeneous version of
$A^{\cyc}_n(x,1, t\,|\,\alpha)$, which is studied   in \cite[Chapter 4]{FS70}. 

For $\s=\sigma_1\ldots \sigma_n\in \S_n$
with  the boundary condition $\mathbf{0-0}$, i.e., 
$\boldsymbol{\sigma_0=\s_{n+1}=0}$, a letter  $\s_i\in[n]$ is called a
  %\begin{itemize}
  % \item 
   {\em valley} (\textbf{val}) of $\s$ if $\sigma_{i-1}>\sigma_{i}<\sigma_{i+1}$;
  %\item  
  {\em peak} (\textbf{pk}) of $\s$ if $\sigma_{i-1}<\sigma_i>\sigma_{i+1}$;
  %\item  
  {\em double ascent} (\textbf{da}) of $\s$ if $\sigma_{i-1}<\sigma_{i}<\sigma_{i+1}$;
  %\item  
  {\em double descent} (\textbf{dd}) of $\s$ if $\sigma_{i-1}>\sigma_{i}>\sigma_{i+1}$.
 % \end{itemize}

It is clear that  the following identities hold
 \begin{align}\label{relation:V and W}
\val=\pk-1,\quad \asc=\val+\lda,\quad \des=\val  +\rdd.
 \end{align} 
Recently,  refining  the  $(\alpha,\beta)$-Eulerian polynomials $A_n(x,y\,|\, \a, \be)$,  
Ji~\cite{Ji23} 
  considered  a variation of Eulerian polynomials incorporating six  statistics over permutations in $\S_{n+1}$:
\begin{equation}\label{refine-CS-JI}
    A_n(u_1,u_2,u_3,u_4\,|\,\alpha, \beta)
:=\sum_{\sigma \in \mathfrak{S}_{n+1}}(u_1u_2)^{{\val}(\sigma)}u_3^{\lda(\sigma)}u_4^{\rdd(\sigma)}{\alpha}^{\lrmax(\sigma)-1}{\beta}^{\rlmax(\sigma)-1}.
\end{equation}
In a follow-up Ji--Lin~\cite{JL23} considered a binomial analogue 
of Carlitz--Scoville's polynomial~\eqref{Carlitz-Scoville-alpha-beta-polynomials} when $\alpha=\beta$.   

This paper originally arose from the desire to provide 
 an alternative approach to  Ji's generating function~\cite[Theorem 1.4]{Ji23} via previous known
 results in the literature,
 which  led us to     
  the following connection formula: let $xy=u_1u_2$ and $x+y=u_3+u_4$, then 
 \begin{equation}\label{CS-Ji connection}
 A_n(u_1,u_2,u_3,u_4\,|\,\alpha, \beta)=A_n^{\cyc}\biggl(x,y,\frac{\a u_3+\be u_4}{\a+\be}\,|\, \a+\be\biggr).
 \end{equation}
By combining  a refined version of \eqref{CS-Ji connection}  and known group actions,  among our main results, we  generalize  Ji and Ji--Lin's polynomials and results in~\cite{Ji23,JL23}.

For  a permutation $\s=\sigma_1\ldots \sigma_n\in \S_n$, we say that an index
 $i\in [n]$ is a
%\begin{itemize}
%\item  
\emph{cycle peak} (\textbf{cpk}) of $\s$ if $\sigma^{-1}(i)<i>\sigma(i)$; 
%\item  
\emph{cycle valley} (\textbf{cval}) of $\s$ if $\sigma^{-1}(i)>i<\sigma(i)$; 
%\item 
\emph{cycle double ascent}  (\textbf{cda}) of $\s$ if $\sigma^{-1}(i)<i<\sigma(i)$; 
%\item  
\emph{cycle double descent}  (\textbf{cdd}) of $\s$ if $\sigma^{-1}(i)>i>\sigma(i)$. 
%\end{itemize}
Note  that $\cpk(\s)=\cval(\s)$.
The following is our  first main result.

\begin{theorem}\label{master1} If $xy=u_1u_2$ and $x+y=u_3+u_4$, then
\begin{equation}\label{eq:master theorem}
A_n^{\cyc}(x,y, t\,|\,\alpha)= 
\sum_{\s\in \S_n} (u_1 u_2)^{\cpk(\s)}u_3^{\cda(\s)}u_4^{\cdd(\s)}t^{\fix(\s)}\alpha^{\cyc(\s)}.
\end{equation}
\end{theorem}

Let  $\sigma=\sigma_1\ldots \sigma_n \in \S_n$ with  the boundary condition $\mathbf{0-0}$.  A letter $\s_i\in[n]$ is a
\begin{itemize}
    \item  \emph{left-to-right-maximum-peak} (\textbf{lmaxpk})  if 
 $\s_i$ is a left-to-right maximum and also a peak;
  \item  \emph{right-to-left-maximum-peak} (\textbf{rmaxpk})  if $\s_i$ is a  right-to-left maximum and also a peak;
\item  \emph{left-to-right-maximum-double-ascent} (\textbf{lmaxda}) if 
 $\s_i$ is a left-to-right  maximum and also  a double ascent;
  \item   \emph{right-to-left-maximum-double-descent} (\textbf{rmaxdd}) if 
   $\s_i$ is a right-to-left maximum and also 
  a double descent.  
  \end{itemize}

Let $\uu=(u_1,u_2,u_3,u_4)$ and define the generalized 
Eulerian polynomial
% \begin{subequations} 
\begin{multline}\label{refine-JI2}
A_n(\uu, f, g,  t\,|\,\alpha, \beta)=\sum_{\sigma \in \mathfrak{S}_{n+1}}(u_1u_2)^{{\val}(\sigma)}u_3^{\lda(\sigma)}u_4^{\rdd(\sigma)}f^{\fmax(\s)-1}
g^{\bmax(\s)-1}\\
\times t^{\lrmaxda(\s)+\rlmaxdd(\s)}
{\alpha}^{{\lrmax}(\sigma)-1}{\beta}^{{\rlmax}(\sigma)-1}.
\end{multline}
  The following  is our second main result, which generalizes \eqref{CS-Ji connection}.

\begin{theorem}\label{master2}
If $xy=u_1u_2$ and $x+y=u_3+u_4$, then
\begin{equation}\label{equ:A(u,s,a,b)-A^cyc(x,y,t,a)-with-s}
A_n(\uu, f, g,  t\,|\,\alpha, \beta)=A_n^{\cyc}\left( x,y, \frac{\a u_3+\beta u_4}{\a f+\beta g}\,t\,|\, \a f+\beta g\right).
\end{equation}
\end{theorem}

In the next section, we present some consequences of the main theorems, and  outline the proofs of \Cref{master1}~and~\ref{master2} in Section~3. 


\section{Applications of the main theorems}  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Exponential  generating functions}
The exponential generating function of polynomials
$ A_n^{\cyc}(x,y, t\,|\,\alpha)$ is well-known~\cite{FS70,CS77,Br00} and reads as follows
% \begin{subequations}
\begin{equation}\label{equ:gen-CS-cyc}
\sum_{n\geq 0} A_n^{\cyc}(x,y, t\,|\,\alpha)\frac{z^n}{n!}=
\left(\frac{(x-y)e^{tz}}{xe^{yz}-ye^{xz}}\right)^\alpha.
\end{equation}

Combining \Cref{master2}  with 
\eqref{equ:gen-CS-cyc}, we  derive immediately the exponential generating function of  $A_n(\uu,f,g,t\,|\,\a,\be)$  in \eqref{refine-JI2}, namely
\begin{theorem}\label{master3} 
Let $xy=u_1u_2$ and  $x+y=u_3+u_4$. We have 
\begin{equation}\label{expgen:P_n-2}
\sum_{n\geq 0}A_{n}(\uu, f,g,t\,|\, {\alpha},{\beta})\frac{z^n}{n!}
 =e^{(\a u_3+\be u_4)tz}\left(\frac{x-y}{xe^{yz}-ye^{xz}}\right)^{\alpha f+\be g}.
 \end{equation}
\end{theorem}

From the above theorem 
we derive plainly the generating functions for 
 Ji's generalized Eulerian polynomials $A_{n}(\uu,1,1,1\,|\, {\alpha},{\beta})$ and Ji--Lin's binomial-Stirling Eulerian polynomials $A_n(\uu, 0, 1,1\,|\, {\alpha},{\beta})$~\cite[Theorem 1.5]{JL23}.
 
 Furthermore, combining \Cref{master2}  and the continued fraction expansion of the 
ordinary generating function of $A_n^{\cyc}(x,y, t\,|\,\alpha)$ in  
\cite[Th\'eor\`eme 3]{Ze93} we obtain  the following continued fraction formula for the ordinary generating function of
$A_{n}(\uu, f,g,t\,|\, {\alpha},{\beta})$.
\begin{theorem}\label{con-frac-A}
We have
\begin{subequations}   
    \begin{equation}
\label{fix-carlitz-scoville-cf}
  \sum_{n=0}^\infty A_n(\uu,f,g,t\,|\,\alpha, \beta)z^n
 = \cfrac{1}{1-b_0 z-\cfrac{\lambda_1z^2}{1-b_1z-\cfrac{\lambda_2 z^2}{1-b_2z-\cdots}}},
\end{equation}
where
\begin{align}
b_k&=k(u_3+u_4)+(\a u_3+\beta u_4)t,\\
\lambda_{k+1}&=(k+\a f+\beta g)(k+1) u_1u_2\quad (k\geq 0).
\end{align}
\end{subequations}
\end{theorem}

In particular, the polynomials $A_{n}(\uu, f,g,t\,|\, {\alpha},{\beta})$ encompass
the moment sequences  of the orthogonal Sheffer polynomials~\cite{Ze21}.

\subsection{\texorpdfstring{$(\alpha,t)$}{(alpha,t)}-Eulerian and binomial-Eulerian polynomials}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Define the   $(\a,t)$-Eulerian polynomials $A_n(x,y, t\,|\,\a)$ by
\begin{subequations}
\begin{align} \label{t-Ji-Lin}   
A_n\left(x,y,t\,|\, \a\right):
=\sum_{\s\in\S_{n+1}}x^{\asc(\s)}y^{\des(\s)} t^{\lrmaxda(\s)+\rlmaxdd(\s)}\a^{\lrmax(\s)+\rlmax(\s)-2},
\end{align}
which is equal to $A_{n}(x, y, x, y, 1,1,t\,|\, \a, \a)$. By \Cref{master2} we have
\begin{equation}\label{An=Ancyc}
    A_n\left(x,y,t\,|\, \a\right)=A_n^{\cyc}\left(x,y,\frac{x+y}{2}t\,|\, 2\a\right).
\end{equation}
Combining \eqref{An=Ancyc} and   
\Cref{master2} with $f=g=1$, $\a=\beta$, we obtain   the following $t$-analogue of   Ji-Lin's Theorem 1.6~\cite{JL23}.
\end{subequations}
\begin{theorem}\label{master2=1}
 If $xy=u_1u_2$ and $x+y=u_3+u_4$, then
\begin{align*}
A_n\left(x,y,t\,|\, \a\right)=\sum_{\sigma \in \mathfrak{S}_{n+1}}(u_1u_2)^{{\val}(\sigma)}u_3^{{\lda}(\sigma)}u_4^{{\rdd}(\sigma)}
t^{\lrmaxda(\s)+\rlmaxdd(\s)}{\alpha}^{{\lrmax}(\sigma)+{\rlmax}(\sigma)-2}. 
\end{align*}
\end{theorem}


We define  the $(\a,t)$-binomial-Eulerian polynomials
$\widetilde{A}_n(x,y,t\,|\,\alpha)$ by
 \begin{subequations}
 \begin{align}
\widetilde{A}_n(x,y,t\,|\,\alpha)
=\sum_{\s\in\PRW_{n+1}}x^{\asc(\s)}y^{\des(\s)}t^{\lrmaxda(\s)+\rlmaxdd(\s)}
\a^{\lrmax(\s)+\rlmax(\s)-2},\label{A-tilde-def}
 \end{align}
 which is equal to $A_{n}(x, y, x, y, 0,1,t\,|\, {\alpha},{\alpha})$ because 
 a permutation $\s\in \S_n$ is an element of $\PRW_n$ if and only if $\fmax(\s)=1$. 
 By  \Cref{master2} we have 
 \begin{equation}
\widetilde{A}_n(x,y,t\,|\,\alpha)=A_n^{\cyc}( x,y,(x+y)t\,|\,\alpha).\label{cyc-lin1}
 \end{equation}
 Combining \eqref{cyc-lin1} and 
 \Cref{master2} 
with    $f=0$,  $g=1$, and $\a=\beta$, we obtain   the following $t$-analogue of   Ji-Lin's Theorem 1.5~\cite{JL23}.
 \begin{theorem}\label{master2=0}
If  $xy=u_1u_2$ and $x+y=u_3+u_4$, then
\begin{align*}
\widetilde{A}_n(x,y,t\,|\,\alpha)=\sum_{\sigma \in \mathfrak{\PRW}_{n+1}}(u_1u_2)^{{\val}(\s)}u_3^{\lda(\sigma)}u_4^{{\rdd}(\sigma)}
t^{\lrmaxda(\s)+\rlmaxdd(\s)}
{\alpha}^{{\lrmax}(\sigma)+{\rlmax}(\sigma)-2}.
\end{align*}
\end{theorem}
 \end{subequations}

\subsection{A symmetric \texorpdfstring{$(\alpha,t)$}{(alpha,t)}-Eulerian identity}\label{sec:Eulerian identity}

Define two kinds of $(\a,t)$-\emph{Eulerian numbers}  as follows:
\begin{subequations}
    \begin{equation}\label{(a,t)-Stirling-Eulerian number}
\left\langle \mycom{n}{k}\right\rangle^{\exc}_{\alpha,t}:=\sum_{
\mycom{\s\in \S_n}{\exc(\s)=n-k}}\alpha^{\cyc(\s)}t^{\fix(\s)}\qquad (1\leq k\leq n),
\end{equation}
and 
\begin{equation}\label{linear-stirling-eulerian-number}
\left\langle \mycom{n}{k}\right\rangle_{\alpha,t}^{\asc}:=
\sum_{\mycom{\s\in \S_n}{\asc(\s)=n-k}}\a^{\rlmax(\s)}
t^{\rlmaxdd(\s)} \qquad ( 1\leq k\leq n).
\end{equation}
\end{subequations}
It is easy to see that $A_n^{\cyc}(x,y,t(x+y)\,|\,\alpha)$  is symmetric in $x$ and $y$ because the 
 involution $\vartheta: \s\mapsto \s^{-1}$ for $\s\in \S_n$ satisfies $(\exc, \drop, \fix)\,\s=(\drop, \exc, \fix)\,\s^{-1}$. We have the following symmetric $(\alpha,t)$-Eulerian identity.

\begin{theorem}\label{sum-equ:bin-(a,t)} For integers $a, b\geq 0$, we have
\begin{subequations}\label{equ-two}
    \begin{align}\label{exc=asc}
    \left\langle \mycom{n}{k}\right\rangle_{\alpha,t}:=\left\langle \mycom{n}{k}\right\rangle_{\alpha,t}^{\exc}=\left\langle \mycom{n}{k}\right\rangle_{\alpha,t}^{\asc}, 
\end{align}
and 
\begin{equation}\label{sym1}
\sum_{k\geq 0}(\alpha t)^{a+b-k} \binom{a+b}{k}
\left\langle \mycom{k}{a}\right\rangle_{\alpha,t}=
\sum_{k\geq 0}(\alpha t)^{a+b-k} \binom{a+b}{k}
\left\langle \mycom{k}{b}\right\rangle_{\alpha,t},
\end{equation}
\end{subequations}
where
 $\left\langle \mycom{0}{k}\right\rangle_{\alpha,t}=\left\langle \mycom{k}{0}\right\rangle_{\alpha,t}=\delta_{k,0}$.
\end{theorem}

\begin{remark}
When $\alpha=t=1$ and $t=1$, identity~\eqref{equ-two} reduces to~\eqref{cgk} and~\cite[Theorem 4.1]{JL23}, respectively.
\end{remark}


\subsection{\texorpdfstring{$\gamma$}{gamma}-positivity of \texorpdfstring{$(\alpha, t)$}{(alpha,t)}-Eulerian polynomials}
The following lemma can be derived from \Cref{master1} and~\ref{master2} with suitable substitutions~\cite[Lemma~2.2]{XZ24}.

\begin{lemma}\label{lemma-gamma-3-cyc-linear} For any variable $f$, we have 
    \begin{subequations}
\label{eq:gamma1}
    \begin{align} 
   A_n&(x,y, 0, x+y, f, 1,  t\,|\,\a, \a)\nonumber\\
    &=\sum_{\substack{\s\in\S_{n+1}\\\da(\s)=0}}(xy)^{\asc(\s)}(x+y)^{n-2\asc(\s)}f^{\fmax(\s)-1}t^{\rlmaxdd(\s)}\a^{\lrmax(\s)+\rlmax(\s)-2}\label{eq:gamma1-a}\\
    &=\sum_{\substack{\s\in\S_n\\\cda(\s)=0}}(xy)^{\exc(\s)}(x+y)^{n-2\exc(\s)}t^{\fix(\s)}\a^{\cyc(\s)}(f+1)^{\cyc(\s)-\fix(\s)}.\label{eq:gamma1b}
    \end{align}
 \end{subequations}
\end{lemma}

We define three types of subsets of $\S_n$:  
\begin{subequations}
    \begin{align}
        &  \S^{\cda=0}_{n, \exc=j}:=\{\s\in \S_n:\cda(\s)=0\,\, {\rm and}\,\,  \exc(\s)=j\};\label{cda=0}\\
        &   \S^{\da=0}_{n,\asc=j}:=\{\s\in \S_n: \da(\s)=0\,\, {\rm and}\,\,  \asc(\s)=j\};\label{da=0}\\
        &   \PRW^{\da=0}_{n,\asc=j}:=\{\s\in \PRW_n: \da(\s)=0\,\, {\rm and}\,\,  \asc(\s)=j\}\label{da=0-PRW}.
    \end{align}
\end{subequations}
From \Cref{lemma-gamma-3-cyc-linear}, we derive the following combinatorial interpretations  of the coefficients in the $\gamma$-expansion of $A_n(x,y,t\,|\,\alpha)$ and $\widetilde{A}_n(x,y,t\,|\,\alpha)$. 

\begin{theorem}
    \label{coro: gamma-expansion-A_n(x,y|a)-cyc-version}
For $0\leq j\leq \lfloor{n/2}\rfloor$, we have

\begin{equation}\label{gamma:A}
A_n(x,y,t\,|\, \a)=\sum_{j=0}^{\lfloor n/2\rfloor}\gamma_{n,j}(\a,t)(xy)^j(x+y)^{n-2j},
\end{equation}
where 
\begin{subequations}\label{gamma-linear1}
 \begin{align}
\gamma_{n,j}(\a,t)&=\sum_{\s\in\S^{\da=0}_{n+1,\asc=j}}\a^{\lrmax(\s)+\rlmax(\s)-2}t^{\rlmaxdd(\s)}\label{gamma-linear1a}\\
&=\sum_{\s\in\S^{\cda=0}_{n,\exc=j}}2^{\cyc(\s)-\fix(\s)}\a^{\cyc(\s)}t^{\fix(\s)};\label{gamma-linear1b}
\end{align}
\end{subequations}
\end{theorem}
\begin{remark}
    When $t=1$, Carlitz and Scoville~\cite{CS74} studied the above $\gamma$-coefficients but did not give any combinatorial interpretation. By complement operation $\s \mapsto \s^c$, Equation~\eqref{gamma-linear1a} with $t=1$ can be obtained in Ji-Lin's~\cite[Theorem~1.6]{JL23}. When $\alpha=t=1$, Equation~\eqref{gamma-linear1a} is equivalent to~\eqref{gamm-r-n-j}.
\end{remark}

\begin{theorem}\label{gamma:P_n}
For $n\ge 1$, we have
\begin{align}
\widetilde{A}_n(x,y,t\,|\,\alpha)
&=\sum_{j=0}^{\lfloor \frac{n}{2}\rfloor}\widetilde{\gamma}_{n,j}(\a,t)(xy)^j(x+y)^{n-2j},\label{equ:gamma:P_n}
\end{align}
where 
\begin{equation}
    \widetilde{\gamma}_{n,j}(\a,t)=\sum_{\s\in\PRW^{\da=0}_{n+1,\asc=j}}\a^{\rlmax(\s)-1}t^{\rlmaxdd(\s)}
=\sum_{\sigma\in  \S^{\cda=0}_{n, \exc=j}}\alpha^{\cyc\,(\sigma)}t^{\fix(\sigma)}.\label{gamma1}
\end{equation}
\end{theorem}

When $t=1$, by  complement operation $\s\mapsto \s^c$, the first combinatorial interpretations in ~\eqref{gamma1}  appeared in~\cite[Theorem 2.1]{JL23}. When $\alpha=t=1$, Equation~\eqref{gamma1} is equivalent to~\eqref{PRW-positivity}.
 More combinatorial interpretations of the $\gamma$-coefficients in~\eqref{gamma:A} and~\eqref{equ:gamma:P_n} are given in \cite[Theorem~2.5 and~2.6]{XZ24}. 

From \Cref{master3}, we can derive the exponential generating function for these two $\gamma$-coefficients, respectively.
\begin{theorem}\label{Thm: egf of gamma coefficients}
 Let $u=\sqrt{1-4x}$. We have
\begin{align}
1+\sum_{n\ge 1}\sum_{j=0}^{\lfloor n/2\rfloor}\gamma_{n,j}(\alpha, t)x^j\frac{z^n}{n!}
&=\biggl(\frac{u\,e^{\frac{1}{2}(t-1)z}}{u\cosh(uz/2)-
\sinh(uz/2) }\biggr)^{2\alpha},\label{equ2:gen-Gamma-coe}\\
1+\sum_{n\ge 1}\sum_{j=0}^{\lfloor n/2\rfloor}\widetilde{\gamma}_{n,j}(\alpha, t)x^j\frac{z^n}{n!}
%&=\biggl(\frac{(x-1)e^{tz}}{xe^{z/(1+x)}-e^{xz/(1+x)}}\biggr)^{\alpha}\nonumber\\
&=\biggl(\frac{u\,e^{(t-\frac{1}{2})z}}{u\cosh(uz/2)-
\sinh(uz/2) }\biggr)^{\alpha}.\label{equ:gen-Gamma-coe}
\end{align} 
\end{theorem}

    

\subsection{\texorpdfstring{$\gamma$}{gamma}-vector of \texorpdfstring{$(\alpha,t)$}{(alpha,t)}-Eulerian polynomials  and cycle André permutations}
%%%%%%
For $0\le j\le \lfloor n/2\rfloor$, let $d_{n,j}(\a,t)=\gamma_{n,j}(\a,t)/2^j$,
then, Equation~\eqref{gamma:A} reads
\begin{equation}\label{gamma:A/2}
A_n(x,y,t\,|\, \a)=\sum_{j=0}^{\lfloor n/2\rfloor}2^jd_{n,j}(\a,t)(xy)^j(x+y)^{n-2j}.
\end{equation} 
From \Cref{con-frac-A}, 
we derive the  continued fraction
    \begin{equation}
  \sum_{n=0}^\infty \sum_{j=0}^{\lfloor n/2\rfloor} d_{n,j}(\a,t) x^j z^n
 = \cfrac{1}{1-b_0 z-\cfrac{\lambda_1z^2}{1-b_1z-\cfrac{\lambda_2 z^2}{1-b_2z-\cdots}}},
\end{equation}
where
\begin{align}
b_k=k+\a t,\quad
\lambda_{k+1}=\binom{k+1}{2}x+\a (k+1) x\quad (k\geq 0).
\end{align}
It follows that  $d_{n,j}(\a,t)$ are  polynomials  in $\N[\a,t]$.   
The aim of this section is to provide three combinatorial interpretations in terms of Andr\'e (resp. cycle Andr\'e) permutations.

\begin{definition}[\cite{HR98}]\label{def:x-factorization-André1-2}
For a fixed $x\in[n]$, let $\s=\s_1\cdots\s_n\in\S_n$. 
Say that $\s$ is an \textbf{André permutation of the first kind (resp. second kind)} 
if $\s$ has no double descents, i.e., $\s_{i-1}>\s_{i}>\s_{i+1}$, and
each factorisation $u\,\lambda(x)\,x\,\rho(x)\,v$ of $\s$ has property
\begin{itemize}
\item $\lambda(x)=\emptyset$ if $\rho(x)=\emptyset$,
\item $\max(\lambda(x))<\max(\rho(x))$  $($resp. $\min(\rho(x))<\min(\lambda(x))$$)$  if  $\lambda(x)\neq\emptyset$,
\end{itemize}
where $\lambda(x)$ and $\rho(x)$ are the the maximal contiguous subword immediately to the left (resp. right) of $x$ whose letters are all greater than $x$.
\end{definition} 

Let $\A_n^1$ (resp. $\A_n^2$) be the set of André permutations of the first (resp. second) kind  in $\S_n$. 
It is known~\cite{FS70} that the cardinality of 
$\A_n^1$ (resp. $\A_n^2$) is the  \emph{Euler number} $E_{n}$.

Let $\s=\s_1\dots\s_n\in\S_n$. A right-to-left minimum (\textbf{rmin}) of $\s$ is an element $\s_i$ such that $\s_j>\s_i$ if $j>i$.  A letter $\s_i\in[n]$ is a right-to-left-minimum-da (\textbf{rminda}) of $\sigma$ if it is a double ascent and $\s_i$ is a rmin. Let $A:=\{a_1,\dots, a_k\}$ be a set of $k$ positive integers. 
Let $C=(a_1,\dots, a_k)$ be a cycle (cyclic permutation) of $A$  with $a_1=\min\{a_1,\dots, a_k\}$. Then,  cycle  $C$ is called an
\textbf{\emph{André cycle}} if the word 
 $a_2\ldots a_k$ is an André permutation of the first kind.
 We say that a  permutation is a \textbf{cycle André permutation}
if it is a product of disjoint André cycles.  Let $\Web_{n}$ be the set of cycle André   permutations  of $[n]$. 
Note that Hwang et al.~\cite{HJO23} used   cycle André permutations  to characterise 
  the so-called \emph{Web permutations} to provide a combinatorial interpretation for entries of the transition matrix between the Specht and $\mathrm{SL}_2$-web bases.


\begin{theorem}\label{main-theorem1}
For $0\le j\le \lfloor n/2\rfloor$, we have 
\begin{subequations}\label{combinatorics of d}
\begin{align}
d_{n,j}(\a,t)=&\sum_{\mycom{\s\in\Web_{n}}{\drop(\s)=j}}t^{\fix(\s)}\a^{\cyc(\s)},\label{combinatorial interpretation of d-web}\\
d_{n,j}(\a,t)=&\sum_{\mycom{\s\in \A_{n+1}^{(i)}}{\des(\s)=j}}t^{\rlminda(\s)}\a^{\rlmin(\s)-1}, \quad (i=1,2)\label{combinatorial interpretation of d-andre}.
\end{align}
\end{subequations}
\end{theorem}

We prove~\eqref{combinatorics of d}
by computing the exponential generating functions of both sides, and derive the $i=1$ case of~\eqref{combinatorial interpretation of d-andre}  from~\eqref{combinatorial interpretation of d-web}  by a bijection from $\Web_n$ to $\A_{n+1}^{1}$, and  the $i=2$ case  by constructing  another bijection from  $\A_{n+1}^1$ to $\A_{n+1}^2$ via   André trees, see~\cite[Section~4.3--4.5]{XZ24}.

\section{Proof outlines of \texorpdfstring{\Cref{master1}~and~\ref{master2}}{the main theorems}}

\subsection{Preliminaries}\label{FFT}
Recall two variants $\theta_1$ and $\theta_2$ of Foata's fundamental transformation~\textbf{FFT}.
For $\s\in \S_n$,  the mapping $\theta_1: \s\mapsto \theta_1(\s)$
(resp. $\theta_2: \s\mapsto \theta_2(\s)$) goes  as follows: (a) Factorize $\s$ as product of disjoint 
cycles with the largest letter in the last (resp. first) position of each cycle; (b) Order the cycles from left to right in decreasing (resp. increasing) order of their largest letters, then erase the parentheses to obtain  $\theta_1(\s)$ (resp. $\theta_2(\s)$). See~\cite[p. 30]{St12}.

Fix a letter $x\in[n]$ and a permutation $\s\in\S_n$, the \emph{x-factorization} of $\s$ is defined as the concatenation $\s=w_1w_2\,x\,w_4w_5$, where $w_2$ (resp. $w_4$) is the maximal contiguous subword immediately to the left (resp. right) of $x$ whose letters are all smaller than $x$. 
Define the involution $\xi_x(\s):\S_n\to\S_n$ by
$$
\xi_x(\s):=
\begin{cases}
w_1w_4\,x\,w_2w_5,& \text{if $x$ is a double ascent or a double descent of $\s$};\\
\s,&\text{if $x$ is a valley  or a peak of $\s$}.\\
\end{cases}
$$
Here, we use the convention $\s(0)=\s(n+1)=\infty$.  
The involution $\xi_S:=\prod_{x\in S}\xi_x$ ($S\subseteq [n]$) defines a $\Z_2^n$ action on $\S_n$~\cite{Br08}, called the \emph{modified Foata and Strehl action} or \emph{MFS} action, see ~\cite{FS74}. Cooper et al.~\cite[ Propositions 3 and 4]{CJZ20} defined the \emph{cyclic valley hopping} $\psi_x(\s):\S_n\to\S_n$ by
$$
\psi_x(\s):=
\begin{cases}
\theta_2^{-1}\circ\xi_x\circ \theta_2(\s),& \text{if $x$ is not a fixed point of $\s$};\\
\s,&\text{if $x$ is a fixed point of $\s$},\\
\end{cases}
$$
where we treat the $0$-th letter of $\theta_2(\s)$ as 0 and the (n+1)-th letter as $\infty$. Define the involution $\psi_S:=\prod_{x\in S}\psi_x (S\subseteq [n])$. 

\subsection{Proof of \texorpdfstring{\Cref{master1}}{Theorem 1.1}} For any permutation $\s\in\S_n$,  denote the orbit of $\s$ under cyclic valley-hopping by $\mathrm{Orb}(\s):=\{\psi_S(\s)|S\subseteq[n]\}$, which has a unique permutation $\bar{\s}$ without cyclic double descents.
By \cite[ Propositions 3 and 4]{CJZ20}, we have 
\begin{subequations}
\begin{equation}\label{Orb:u}
\sum_{\pi\in\mathrm{Orb}(\s)}(u_1u_2)^{\cpk(\pi)}u_3^{\cda(\pi)}u_4^{\cdd(\pi)}t^{\fix(\pi)}\a^{\cyc(\pi)}=
(u_1u_2)^{\cpk(\bar{\s})}(u_3+u_4)^{\cda(\bar{\s})}t^{\fix(\bar{\s})}\a^{\cyc(\bar{\s})}.
\end{equation}

By definition, it is clear that for $\s\in\S_n$,
\begin{equation}\label{exc-drop}
   \exc(\s)=\cda(\s)+\cpk(\s)\quad \text{and} \quad \drop(\s)=\cdd(\s)+\cval(\s). 
\end{equation}
Setting $u_1=u_3=x$ and $u_2=u_4=y$ in~\eqref{Orb:u} yields
\begin{equation}\label{Orb:x}
\sum_{\pi\in\mathrm{Orb}(\s)}x^{\exc(\pi)}y^{\drop(\pi)}t^{\fix(\pi)}\a^{\cyc(\pi)}=(xy)^{\cpk(\bar{\s})}(x+y)^{\cda(\bar{\s})}t^{\fix(\bar{\s})}\a^{\cyc(\bar{\s})}.
\end{equation}
Thus, if $u_1u_2=xy$ and $u_3+u_4=x+y$, combining 
\eqref{Orb:u} and \eqref{Orb:x} we have 
\[
\sum_{\pi\in\mathrm{Orb}(\s)}(u_1u_2)^{\cpk(\pi)}u_3^{\cda(\pi)}u_4^{\cdd(\pi)}t^{\fix(\pi)}\a^{\cyc(\pi)}=\sum_{\pi\in\mathrm{Orb}(\s)}x^{\exc(\pi)}y^{\drop(\pi)}t^{\fix(\pi)}\a^{\cyc(\pi)}. 
\]
\end{subequations}
 Then summing over all the orbits of $\S_n$ gives Equation~\eqref{eq:master theorem}.\qed

\subsection{Proof of \texorpdfstring{\Cref{master2}}{Theorem 1.2}}
For a finite set of positive integers  $E$,  we denote by $\S_E$ 
the set of permutations of $E$.  For   $\s\in \S_E$  define the weight function
\begin{subequations}
\begin{align}
&w(\s; \a,a, b)=(u_1 u_2)^{\cpk(\s)}u_3^{\cda(\s)}u_4^{\cdd(\s)} a^{\fix(\s)}b^{\cyc(\s)-\fix(\s)}\a^{\cyc(\s)},\\
&w_1(\s; \a,a, b,t)=w(\s; \a,a, b)\,t^{\lrmaxda(\s)},\\
&w_2(\tau; \a,a, b,t)=w(\s; \a,a, b)\,t^{\rlmaxdd(\tau)}.
\end{align}
\end{subequations}
Let $\S_n^{\bigstar}$ be the set of permutations in $\S_n$ of which each cycle has a color in $\{\red{Red}, \blue{Blue}\}$. An element of $\S_n^{\bigstar}$ is called a \emph{cycle decorated permutation} (CDP).
Hence,  a CDP  $\pi\in \S_n^{\bigstar}$ is in bijection 
with a pair of permutations 
 $(\s, \tau)\in \S_A\times \S_B$ such that $(A,B)$ is an ordered set partition of $[n]$ and $\pi=\s \tau$, namely,
the permutation $\s$ consists of 
all red cycles and $\tau$ the blue ones. 
By \Cref{master1}, under the assumption that $xy=u_1u_2$ and  $x+y=u_3+u_4$, 
the left-hand side of equality \eqref{equ:A(u,s,a,b)-A^cyc(x,y,t,a)-with-s} can be written as 
\begin{align}\label{equ:A_n^cyc with weight function}
A_n^{\cyc}&\left( x,y, t(\a u_3+\beta u_4)(\a f+\beta g)^{-1}\,|\, \a f+\beta g\right)\nonumber\\
&=\sum_{\pi\in\S_n}(u_1 u_2)^{\cpk(\pi)}u_3^{\cda(\pi)}u_4^{\cdd(\pi)}(t(\a u_3+\be u_4))^{\fix(\pi)}(\alpha f+\be g)^{\cyc(\pi)-\fix(\pi)}\nonumber\\
&=\sum_{(\s,\tau)\in \S_n^{\bigstar}}w_1(\s; \a, u_3, f ,t)\,w_2(\tau; \beta, u_4, g,t).
\end{align}
We define a mapping  $\rho:\S_n^{\bigstar}\to  \S_{n+1}$ as follows:
\begin{align}\label{mapping: rho}
(\s,\tau)\mapsto \widetilde{\pi}:=\theta_2(\s)\,x\, \theta_1(\tau) \quad \text{with}\,\,  x=n+1,
\end{align}
where $\theta_1$ and $\theta_2$ are the FFT, see~\Cref{FFT}. Clearly this is a bijection. It follows \eqref{equ:A(u,s,a,b)-A^cyc(x,y,t,a)-with-s} from combining $\rho$ and~\eqref{equ:A_n^cyc with weight function}.







%\acknowledgements{\lipsum[20]}
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\end{document}
