\documentclass[12pt,reqno]{article} \usepackage[usenames]{color} \usepackage{amssymb} \usepackage{graphicx} \usepackage{amscd} \usepackage[colorlinks=true, linkcolor=webgreen, filecolor=webbrown, citecolor=webgreen]{hyperref} \definecolor{webgreen}{rgb}{0,.5,0} \definecolor{webbrown}{rgb}{.6,0,0} \usepackage{color} \usepackage{fullpage} \usepackage{float} \usepackage{psfig} \usepackage{graphics,amsmath,amssymb} \usepackage{amsthm} \usepackage{amsfonts} \usepackage{latexsym} \usepackage{epsf} \setlength{\textwidth}{6.5in} \setlength{\oddsidemargin}{.1in} \setlength{\evensidemargin}{.1in} \setlength{\topmargin}{-.1in} \setlength{\textheight}{8.4in} \newcommand{\seqnum}[1]{\href{http://oeis.org/#1}{\underline{#1}}} \begin{document} \begin{center} \epsfxsize=4in \leavevmode\epsffile{logo129.eps} \end{center} \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{conjecture}[theorem]{Conjecture} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \begin{center} \vskip 1cm{\LARGE\bf A New Class of Refined Eulerian Polynomials } \vskip 1cm \large Hua Sun\thanks{Partially supported by the National Natural Science Foundation of China (Grant No.~11526044) and the Doctoral Scientific Research Starting Foundation of Liaoning Province (No.~20170520451).} \\ College of Sciences \\ Dalian Ocean University \\ Dalian 116023 \\ P. R. China \\ \href{mailto:sunhua@dlou.edu.cn}{\tt sunhua@dlou.edu.cn} \end{center} \vskip .2 in \newcommand{\cs}{{\mathfrak{S}}} \def \odd{{\rm odd}\,} \def \des{{\rm des}\,} \def \odes{{\rm odes}\,} \def \edes{{\rm edes}\,} \def \oasc{{\rm oasc}\,} \def \easc{{\rm easc}\,} \def \asc{{\rm asc}\,} \def \Odes{{\rm Odes}\,} \def \Edes{{\rm Edes}\,} \def \Oasc{{\rm Oasc}\,} \def \Easc{{\rm Easc}\,} \def \Des{{\rm Des}\,} \def \Asc{{\rm Asc}\,} \def \Orb{{\rm Orb}\,} \newcommand{\lrf}[1]{\left\lfloor #1\right\rfloor} \newcommand{\lrc}[1]{\left\lceil #1\right\rceil} \begin{abstract} In this note we introduce a new class of refined Eulerian polynomials defined by $$A_n(p,q)=\sum_{\pi\in\cs_{n}}p^{\odes(\pi)}q^{\edes(\pi)},$$ where $\odes(\pi)$ and $\edes(\pi)$ enumerate the number of descents of permutation $\pi$ in odd and even positions, respectively. We show that the refined Eulerian polynomials $A_{2k+1}(p,q),k=0,1,2,\ldots,$ and $(1+q)A_{2k}(p,q),k=1,2,\ldots,$ have a nice symmetry property. \end{abstract} \section{Introduction} Let $f(q)=a_{r}q^r+\cdots+a_{s}q^s(r\leq s)$, with $a_{r}\neq 0$ and $a_{s}\neq 0$, be a real polynomial. The polynomial $f(q)$ is {\it palindromic} if $a_{r+i}=a_{s-i}$ for any $i$. Following Zeilberger~\cite{Zei88}, define the {\it darga} of $f(q)$ to be $r+s$. The set of all palindromic polynomials of darga $n$ is a vector space~\cite{SWZ15} with gamma basis $$\Gamma_{n}:=\{q^{i}(1+q)^{n-2i} \mid 0\leq i\leq \lrf{n/2}\}.$$ Let $f(p,q)$ be a nonzero bivariate polynomial. The polynomial $f(p,q)$ is {\it palindromic of darga $n$} if it satisfies the following two equations: \begin{gather*} f(p,q)=f(q,p),\\ f(p,q)=(pq)^{n}f(1/p,1/q). \end{gather*} See~Adin et al.~\cite{ABERS17} for details. It is known~\cite{Lin16} that the set of all palindromic bivariate polynomials of darga $n$ is a vector space with gamma basis $$\mathcal{B}_{n}:=\{(pq)^{i}(p+q)^{j}(1+pq)^{n-2i-j} \mid i,j\geq0,2i+j\leq n\}.$$ Let $\cs_{n}$ denote the set of all permutations of the set $[n]:=\{1,2,\ldots,n\}$. For a permutation $\pi=\pi_{1}\pi_{2}\cdots\pi_{n}\in\cs_n$, an index $i\in[n-1]$ is a {\it descent} of $\pi$ if $\pi_i>\pi_{i+1}$, and $\des(\pi)$ denotes the number of descents of $\pi$. The classic Eulerian polynomial is defined as the generating polynomial for the statistic des over the set $\cs_{n}$, i.e., $$A_{n}(q)=\sum_{\pi\in\cs_{n}}q^{\des(\pi)}.$$ Foata and Sch\"{u}tzenberger~\cite{FS70} proved that the Eulerian polynomial $A_{n}(q)$ can be expressed in terms of the gamma basis $\Gamma_{n}$ with nonnegative integer coefficients. A polynomial with nonnegative coefficients under the gamma basis $\Gamma_{n}$ is palindromic and unimodal~\cite{Pet15}. Ehrenborg and Readdy~\cite{ER16} studied the number of ascents in odd position on $0,1$-words. We define similar statistics on permutations. For a permutation $\pi\in\cs_{n}$, an index $i\in[n-1]$ is an {\it odd descent} of $\pi$ if $\pi_i>\pi_{i+1}$ and $i$ is odd, an {\it even descent} of $\pi$ if $\pi_i>\pi_{i+1}$ and $i$ is even, an {\it odd ascent} of $\pi$ if $\pi_i<\pi_{i+1}$ and $i$ is odd, an {\it even ascent} of $\pi$ if $\pi_i<\pi_{i+1}$ and $i$ is even. Let $\Odes(\pi)$, $\Edes(\pi)$, $\Oasc(\pi)$ and $\Easc(\pi)$ denote the set of all odd descents, even descents, odd ascents and even ascents of $\pi$, respectively. The corresponding cardinalities are $\odes(\pi)$, $\edes(\pi)$, $\oasc(\pi)$ and $\easc(\pi)$, respectively. Note that we can also define the above four statistics on words of length $n$. The joint distribution of odd and even descents on $\cs_n$ is denoted by $A_n(p,q)$, i.e., $$A_n(p,q)=\sum_{\pi\in\cs_{n}}p^{\odes(\pi)}q^{\edes(\pi)}.$$ The polynomial $A_n(p,q)$ is a bivariate polynomial of degree $n-1$. The monomial with degree $n-1$ is $p^{\lrf{n/2}}q^{\lrf{(n-1)/2}}$ only. If $p=q$, then $A_n(q,q)=A_n(q)$ is the classic Eulerian polynomial. Thus $A_n(p,q)$, $n=1,2,\ldots,$ can be seen as a class of refined Eulerian polynomials. For example, we have \begin{align*} A_1(p,q)&=1,\\ A_2(p,q)&=1+p,\\ A_3(p,q)&=1+2p+2q+pq,\\ A_4(p,q)&=1+6p+5q+5p^2+6pq+p^{2}q,\\ A_5(p,q)&=1+13p+13q+16p^2+34pq+16q^2+13p^{2}q+13pq^{2}+p^{2}q^{2},\\ A_6(p,q)&=1+29p+28q+89p^2+152pq+61q^2+61p^{3}+152p^{2}q\\ &~~~+89pq^{2}+28p^{3}q+29p^{2}q^{2}+p^{3}q^{2}. \end{align*} For convenience, we denote \begin{equation*} \widetilde{A}_n(p,q)= \begin{cases} A_n(p,q), & \text{if $n=2k+1$,}\\ (1+q)A_n(p,q), & \text{if $n=2k$.} \end{cases} \end{equation*} Our main result is the following \begin{theorem}\label{pal} For any $n=1,2,\ldots,$ the polynomial $\widetilde{A}_{n}(p,q)$ is palindromic of darga $\lrf{\frac{n}{2}}$. \end{theorem} In the next section we give a proof of Theorem~\ref{pal}. In Section 3 we study the case $q=1$ and the case $p=1$, the polynomials $A_n(p,1)$ and $A_n(1,q)$ are the generating functions for the statistics odes and edes over the set $\cs_n$, respectively. In the last section, we propose a conjecture that $\widetilde{A}_{n}(p,q)$ can be expressed in terms of the gamma basis $\mathcal{B}_{\lrf{\frac{n}{2}}}$ with nonnegative integer coefficients. \section{The proof of Theorem~\ref{pal}} Let $\pi=\pi_{1}\pi_{2}\cdots\pi_{n}\in\cs_n$, we define the {\it reversal} $\pi^r$ of $\pi$ to be $$\pi^r:=\pi_{n}\pi_{n-1}\cdots\pi_{1},$$ the {\it complement} $\pi^c$ of $\pi$ to be $$\pi^c:=(n+1-\pi_{1})(n+1-\pi_{2})\cdots(n+1-\pi_{n}),$$ and the {\it reversal-complement} $\pi^{rc}$ of $\pi$ to be $$\pi^{rc}:=(\pi^{c})^r=(\pi^{r})^c.$$ If $i$ is a descent of $\pi$, then $i$ is an ascent of $\pi^c$ and if $i$ is an ascent of $\pi$, then $i$ is a descent of $\pi^c$. In other words, $\odes(\pi)+\odes(\pi^c)=\lrf{\frac{n}{2}}$ and $\edes(\pi)+\edes(\pi^c)=\lrf{\frac{n-1}{2}}$. Then \begin{equation*} \begin{split} A_{n}(p,q)&=\sum_{\pi\in\cs_{n}}p^{\odes(\pi)}q^{\edes(\pi)}=\sum_{\pi\in\cs_{n}}p^{\lrf{\frac{n}{2}}-\odes(\pi^c)}q^{\lrf{\frac{n-1}{2}}-\edes(\pi^c)}\\ &=p^{\lrf{\frac{n}{2}}}q^{\lrf{\frac{n-1}{2}}}\sum_{\pi\in\cs_{n}}\left(\frac{1}{p}\right)^{\odes(\pi^c)}\left(\frac{1}{q}\right)^{\edes(\pi^c)} =p^{\lrf{\frac{n}{2}}}q^{\lrf{\frac{n-1}{2}}}A_{n}\left(\frac{1}{p},\frac{1}{q}\right). \end{split} \end{equation*} Specially, for any $k=1,2,\ldots$, we have $A_{2k}(p,q)=p^{k}q^{k-1}A_{2k}(1/p,1/q)$ and for any $k=0,1,2,\ldots$, we have $A_{2k+1}(p,q)=(pq)^{k}A_{2k+1}(1/p,1/q)$. It can be derived that $i$ is a descent of $\pi$ if and only if $i$ is an ascent of $\pi^c$. It is also easy to see that $i$ is a descent of $\pi$ if and only if $n-i$ is an ascent of $\pi^r$. Then, given a permutation $\pi=\pi_{1}\pi_{2}\cdots\pi_{2k+1}\in\cs_{2k+1}$, \begin{center} $i$ is a descent of $\pi$ if and only if $2k+1-i$ is a descent of $\pi^{rc}$. \end{center} Specially, $i$ is an odd descent of $\pi$ if and only if $2k+1-i$ is an even descent of $\pi^{rc}$, and $i$ is an even descent of $\pi$ if and only if $2k+1-i$ is an odd descent of $\pi^{rc}$. So we have $$\widetilde{A}_{2k+1}(p,q)=\sum_{\pi\in\cs_{2k+1}}p^{\odes(\pi)}q^{\edes(\pi)}=\sum_{\pi\in\cs_{2k+1}}p^{\edes(\pi^{rc})}q^{\odes(\pi^{rc})}=\widetilde{A}_{2k+1}(q,p).$$ Thus for any $k=1,2,\ldots,$ the polynomial $\widetilde{A}_{2k+1}(p,q)$ is palindromic of darga $k$. In addition, $$\widetilde{A}_{2k}(p,q)=(1+q)p^{k}q^{k-1}A_{2k}\left(\frac{1}{p},\frac{1}{q}\right)=\left(1+\frac{1}{q}\right)p^{k}q^{k}A_{2k}\left(\frac{1}{p},\frac{1}{q}\right) =(pq)^{k}\widetilde{A}_{2k+1}\left(\frac{1}{p},\frac{1}{q}\right).$$ The last part is to prove that $\widetilde{A}_{2k}(p,q)=\widetilde{A}_{2k}(q,p)$, that is, $$\sum_{\pi\in\cs_{2k}}p^{\odes(\pi)}[q^{\edes(\pi)}+q^{\edes(\pi)+1}]=\sum_{\pi\in\cs_{2k}}q^{\odes(\pi)}[p^{\edes(\pi)}+p^{\edes(\pi)+1}].$$ Let $\cs'_{2k}=\left\{\pi(2k+1),\pi0 \mid \pi\in\cs_{2k}\right\}$, $\cs''_{2k}=\left\{(2k+1)\pi,0\pi \mid \pi\in\cs_{2k}\right\}$, and let $\pi=\pi_{1}\pi_{2}\cdots\pi_{2k}\in\cs_{2k}$. Define a map $\psi:\cs'_{2k}\rightarrow\cs''_{2k}$ by \begin{equation*} \psi(\pi x)= \begin{cases} (2k+1)(2k+1-\pi_{2k})(2k+1-\pi_{2k-1})\cdots(2k+1-\pi_{1}), & \text{if $x=0$,}\\ 0(2k+1-\pi_{2k})(2k+1-\pi_{2k-1})\cdots(2k+1-\pi_{1}), & \text{if $x=2k+1$.} \end{cases} \end{equation*} Given a permutation $\pi\in\cs_{2k}$, it is no hard to see that \begin{align*} &\odes(\pi(2k+1))=\odes(\pi),&\edes(\pi(2k+1))=\edes(\pi),\\ &\odes(\pi0)=\odes(\pi),&\edes(\pi0)=\edes(\pi)+1,\\ &\odes((2k+1)\pi)=\edes(\pi)+1,&\edes((2k+1)\pi)=\odes(\pi),\\ &\odes(0\pi)=\edes(\pi),&\edes(0\pi)=\odes(\pi). \end{align*} Thus \begin{align*} &\odes(\psi(\pi(2k+1)))=\odes(0\pi^{rc})=\edes(\pi^{rc}),\\ &\edes(\psi(\pi(2k+1)))=\edes(0\pi^{rc})=\odes(\pi^{rc}),\\ &\odes(\psi(\pi0))=\odes((2k+1)\pi^{rc})=\edes(\pi^{rc})+1,\\ &\edes(\psi(\pi0))=\edes((2k+1)\pi^{rc})=\odes(\pi^{rc}). \end{align*} Obviously, the map $\psi$ is an involution. Then \begin{equation*} \begin{split} &\sum_{\pi\in\cs_{2k}}p^{\odes(\pi)}[q^{\edes(\pi)}+q^{\edes(\pi)+1}]\\ &=\sum_{\pi\in\cs_{2k}}p^{\odes(\pi(2k+1))}q^{\edes(\pi(2k+1))}+\sum_{\pi\in\cs_{2k}}p^{\odes(\pi0)}q^{\edes(\pi0)}\\ &=\sum_{\pi\in\cs_{2k}}p^{\odes(\psi(\pi(2k+1)))}q^{\edes(\psi(\pi(2k+1)))}+\sum_{\pi\in\cs_{2k}}p^{\odes(\psi(\pi0))}q^{\edes(\psi(\pi0))}\\ &=\sum_{\pi\in\cs_{2k}}p^{\edes(\pi^{rc})}q^{\odes(\pi^{rc})}+\sum_{\pi\in\cs_{2k}}p^{\edes(\pi^{rc})+1}q^{\odes(\pi^{rc})}\\ &=\sum_{\pi\in\cs_{2k}}q^{\odes(\pi)}[p^{\edes(\pi)}+p^{\edes(\pi)+1}]. \end{split} \end{equation*} Thus for any $k=1,2,\ldots,$ the polynomial $\widetilde{A}_{2k}(p,q)$ is palindromic of darga $k$. This completes the proof. \section{The case $p=1$ and the case $q=1$} If $q=1$, the polynomial $A_n(p,1)$ is the generating function for the statistic odes over the set $\cs_n$, and if $p=1$, the polynomial $A_n(1,q)$ is the generating function for the statistic edes over the set $\cs_n$. More precisely, we have \begin{proposition} Let $n$ be a positive integer. Then \begin{equation}\label{equ1} \sum_{\pi\in\cs_{n}}p^{\odes(\pi)}=A_n(p,1)=\frac{n!}{2^{\lrf{\frac{n}{2}}}}(1+p)^{\lrf{\frac{n}{2}}}, \end{equation} and \begin{equation}\label{equ2} \sum_{\pi\in\cs_{n}}q^{\edes(\pi)}=A_n(1,q)=\frac{n!}{2^{\lrf{\frac{n-1}{2}}}}(1+q)^{\lrf{\frac{n-1}{2}}}. \end{equation} \end{proposition} \begin{proof} It is easy to verify that the equalities~\ref{equ1} and~\ref{equ1} are true for $n=1$ and $n=2$. Let $n\geq3$ and let $\pi=\pi_{1}\pi_{2}\cdots\pi_{n}\in\cs_n$. For any $i=1,2,\ldots,\lrf{n/2}$, define a map $\varphi_{i}:\cs_{n}\rightarrow\cs_{n}$ by $$\varphi_{i}(\pi)=\pi_{1}\pi_{2}\cdots\pi_{2i}\pi_{2i-1}\cdots\pi_{n},$$ i.e., $\varphi_{i}(\pi)$ is obtained by swapping $\pi_{2i}$ with $\pi_{2i-1}$ in $\pi$. Obviously, the map $\varphi_{i}$ is an involution, $i=1,2,\ldots,\lrf{n/2}$, and $\varphi_{i}$ and $\varphi_{j}$ commute for all $i,j\in\{1,2,\ldots,\lrf{n/2}\}$. For any subset $S\subseteq \{1,2,\ldots,\lrf{n/2}\}$, we define a map $\varphi_{S}:\cs_{n}\rightarrow\cs_{n}$ by $$\varphi_{S}(\pi)=\prod_{i\in S}\varphi_{i}(\pi).$$ The group $\mathbb{Z}_{2}^{\lrf{n/2}}$ acts on $\cs_{n}$ via the maps $\varphi_{S},S\subseteq \{1,2,\ldots,\lrf{n/2}\}$. For any $\pi\in\cs_n$, let $\Orb^*(\pi)$ denote the orbit including $\pi$ under the group action. There is a unique permutation in $\Orb^*(\pi)$, denoted by $\hat{\pi}$, such that $$\hat{\pi}_{1}<\hat{\pi}_{2},~ \hat{\pi}_{3}<\hat{\pi}_{4},~ \ldots,~ \hat{\pi}_{2\lrf{n/2}-1}<\hat{\pi}_{2\lrf{n/2}}.$$ It is not hard to prove that $\odes(\hat{\pi})=0$ and $\odes(\varphi_{S}(\hat{\pi}))=|S|$ for any $S\subseteq \{1,2,\ldots,\lrf{n/2}\}$. Then $$\sum_{\sigma\in \Orb^*(\pi)}p^{\odes(\sigma)}=(1+p)^{\lrf{\frac{n}{2}}}.$$ Let $\cs^*_{n}$ consist of all the permutations in $\cs_{n}$ such that $$\pi_{1}<\pi_{2},~ \pi_{3}<\pi_{4},~ \ldots,~ \pi_{2\lrf{n/2}-1}<\pi_{2\lrf{n/2}}.$$ The cardinality of the set $\cs^*_{n}$ is $$\binom{n}{2}\binom{n-2}{2}\cdots\binom{n+2-2\lrf{\frac{n}{2}}}{2}=\frac{n!}{2^{\lrf{\frac{n}{2}}}}.$$ Then $$\sum_{\pi\in\cs_{n}}p^{\odes{(\pi)}}=A_n(p,1)=\frac{n!}{2^{\lrf{\frac{n}{2}}}}(1+p)^{\lrf{\frac{n}{2}}}.$$ Similarly, for any $i=1,2,\ldots,\lrf{(n-1)/2}$, we define a map $\phi_{i}:\cs_{n}\rightarrow\cs_{n}$ by $$\phi_{i}(\pi)=\pi_{1}\cdots\pi_{2i+1}\pi_{2i}\cdots\pi_{n},$$ i.e., $\phi_{i}(\pi)$ is obtained by swapping $\pi_{2i}$ with $\pi_{2i+1}$ in $\pi$. Obviously, the map $\phi_{i}$ is an involution, $i=1,2,\ldots,\lrf{(n-1)/2}$, and $\phi_{i}$ and $\phi_{j}$ commute for all $i,j\in\{1,2,\ldots,\lrf{(n-1)/2}\}$. For any subset $S\subseteq \{1,2,\ldots,\lrf{(n-1)/2}\}$, we define a map $\phi_{S}:\cs_{n}\rightarrow\cs_{n}$ by $$\phi_{S}(\pi)=\prod_{i\in S}\phi_{i}(\pi).$$ The group $\mathbb{Z}_{2}^{\lrf{(n-1)/2}}$ acts on $\cs_{n}$ via the maps $\phi_{S},S\in [\lrf{(n-1)/2}]$. For any $\pi\in\cs_n$, let $\Orb^{**}(\pi)$ denote the orbit including $\pi$ under the group action. There is a unique permutation in $\Orb^{**}(\pi)$, denoted by $\bar{\pi}$, such that $$\bar{\pi}_{2}<\bar{\pi}_{3},~ \bar{\pi}_{4}<\bar{\pi}_{5},~ \ldots,~ \bar{\pi}_{2\lrf{(n-1)/2}}<\bar{\pi}_{2\lrf{(n-1)/2}+1}.$$ It is easily obtained that $\edes(\bar{\pi})=0$ and $\edes(\phi_{S}(\bar{\pi}))=|S|$ for any $S\subseteq \{1,2,\ldots,\\ \lrf{(n-1)/2}\}$. Then $$\sum_{\sigma\in \Orb^{**}(\pi)}q^{\edes(\sigma)}=(1+q)^{\lrf{\frac{n-1}{2}}}.$$ Let $\cs^{**}_{n}$ consist of all the permutations in $\cs_{n}$ such that $$\pi_{2}<\pi_{3},~ \pi_{4}<\pi_{5},~ \ldots,~ \pi_{2\lrf{(n-1)/2}}<\pi_{2\lrf{(n-1)/2}+1}.$$ The cardinality of the set $\cs^{**}_{n}$ is \begin{equation*} \begin{cases} \binom{n}{2}\binom{n-2}{2}\cdots\binom{n+2-2\lrf{\frac{n-1}{2}}}{2}=\frac{n!}{2^{\lrf{\frac{n-1}{2}}}}, & \text{if $n$ is odd,}\\ 2\binom{n}{2}\binom{n-2}{2}\cdots\binom{n+2-2\lrf{\frac{n-1}{2}}}{2}=\frac{n!}{2^{\lrf{\frac{n-1}{2}}}}, & \text{if $n$ is even.} \end{cases} \end{equation*} Then $$\sum_{\pi\in\cs_{n}}q^{\edes(\pi)}=A_n(1,q)=\frac{n!}{2^{\lrf{\frac{n-1}{2}}}}(1+q)^{\lrf{\frac{n-1}{2}}}.$$ \end{proof} \section{Remarks} The set of palindromic bivariate polynomials of darga $k$ is a vector space with gamma basis $$\mathcal{B}_{k}=\{(pq)^{i}(p+q)^{j}(1+pq)^{k-2i-j} \mid i,j\geq0,2i+j\leq k\}.$$ Thus the refined Eulerian polynomials $\widetilde{A}_n(p,q)$, $n=1,2,\ldots,$ can be expanded in terms of the gamma basis $\mathcal{B}_{\lrf{\frac{n}{2}}}$. For example, \begin{align*} \widetilde{A}_{1}(p,q)&=A_{1}(p,q)=1,\\ \widetilde{A}_{2}(p,q)&=(1+q)A_{2}(p,q)=(1+q)(1+p)=1+p+q+pq\\ &=(1+pq)+(p+q),\\ \widetilde{A}_3(p,q)&=A_{3}(p,q)=1+2p+2q+pq=(1+pq)+2(p+q),\\ \widetilde{A}_{4}(p,q)&=(1+q)A_{4}(p,q)=(1+q)(1+6p+5q+5p^2+6pq+p^{2}q)\\ &=1+6p+6q+5p^2+12pq+5q^2+6p^{2}q+6pq^{2}+p^{2}q^{2}\\ &=(1+pq)^2+6(p+q)(1+pq)+5(p+q)^2,\\ \widetilde{A}_{5}(p,q)&=A_{5}(p,q)=1+13p+13q+16p^2+34pq+16q^2+13p^{2}q+13pq^{2}+p^{2}q^{2}\\ &=(1+pq)^2+13(p+q)(1+pq)+16(p+q)^2,\\ \widetilde{A}_{6}(p,q)&=(1+q)A_{6}(p,q)\\ &=(1+q)(1+29p+28q+89p^2+152pq+61q^2\\ &~~~+61p^{3}+152p^{2}q+89pq^{2}+28p^{3}q+29p^{2}q^{2}+p^{3}q^{2})\\ &=1+29p+29q+89p^2+89q^2+181pq+61p^3+241p^{2}q\\ &~~~+241pq^{2}+61q^3+181p^{2}q^{2}+89p^{3}q+89pq^{3}+29p^{3}q^{2}+29p^{2}q^{3}+p^{3}q^{3}\\ &=(1+pq)^3+29(p+q)(1+pq)^2+89(p+q)^2(1+pq)+61(p+q)^3. \end{align*} We conjecture that for any $n\geq 1$, all $c_{j}$ are positive integers in the following expansion $$\widetilde{A}_n(p,q)=\sum_{j=0}^{\lrf{\frac{n}{2}}}c_{j}(p+q)^{j}(1+pq)^{\lrf{\frac{n}{2}}-j}.$$ \section{Acknowledgment} I am grateful to my advisor Prof.~Yi Wang for his valuable comments and suggestions. I also would like to thank the referee for his/her careful reading and many helpful suggestions. \begin{thebibliography}{9} \bibitem{ABERS17}R. M. Adin, E. Bagno, E. Eisenberg, S. Reches, and M. Sigron, \textrm{Towards a combinatorial proof of Gessel's conjecture on two-sided gamma positivity: a reduction to simple permutations,} preprint, 2017. Available at \url{http://arxiv.org/abs/1711.06511}. \bibitem{ER16}R. Ehrenborg and M. A. Readdy, \textrm{The Gaussian coefficient revisited,} \textit{J. Integer Sequences,} \textbf{19} (2016), \href{https://cs.uwaterloo.ca/journals/JIS/VOL19/Ehrenborg/ehr3.html}{Article 16.7.8}. \bibitem{FS70}D. Foata and M.-P. Sch\"{u}tzenberger, \textit{Th\'{e}orie G\'{e}om\'{e}trique des Polyn\^{o}mes Eul\'{e}riens,} Lecture Notes in Mathematics, Vol.~138, Springer-Verlag, 1970. \bibitem{Lin16}Z. Lin, \textrm{Proof of Gessel's $\gamma$-positivity conjecture,} \textit{Electron. J. Combin.,} \textbf{23} (3) (2016), paper P3.15. \bibitem{Pet15}T. K. Petersen, \textit{Eulerian Numbers,} Birkhauser, 2015. \bibitem{SWZ15}H. Sun, Y. Wang, and H. X. Zhang, \textrm{Polynomials with palindromic and unimodal coefficients,} \textit{Acta Mathematica Sinica, English Series,} \textbf{31} (4) (2015), 565--575. \bibitem{Zei88}D. Zeilberger, \textrm{A one-line high school proof of the unimodality of the Gaussian polynomials $\binom{n}{k}_q$ for $k<20$,} in D. Stanton, ed., \textit{$q$-Series and Partitions}, IMA Volumes in Mathematics and Its Applications, Vol.~18, Springer, 1989, pp.~67--72. \end{thebibliography} \bigskip \hrule \bigskip \noindent 2010 {\it Mathematics Subject Classification}: Primary 05A05; Secondary 05A15, 05A19. \noindent \emph{Keywords: } odd descent, even descent, Eulerian polynomial, $\gamma$-positivity. \bigskip \hrule \bigskip \vspace*{+.1in} \noindent Received January 31 2018; revised versions received May 13 2018; May 17 2018. Published in {\it Journal of Integer Sequences}, May 26 2018. \bigskip \hrule \bigskip \noindent Return to \htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}. \vskip .1in \end{document} .