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latexsym, amscd, epsfig, epic,color}
\usepackage{mathtools} 
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\usepackage{eucal}%    caligraphic-euler fonts: \mathcal{ }
\usepackage{eufrak}%   frak-euler        fonts: \mathfrak{ }
%\usepackage[all]{xypic}
\usepackage{xspace}
\usepackage{a4wide}
\usepackage{colordvi}
\usepackage{comment}
\usepackage{hyperref,eucal}
\usepackage{tikz}\usetikzlibrary{matrix,arrows}

%\newtheorem{thm}{Theorem}
%\newtheorem{lem}{Lemma}

%\usepackage{lipsum}

%\newcounter{mycount}
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem*{main}{Main~Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{conjecture}[theorem]{Conjecture}
\theoremstyle{definition}
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%\newtheorem{oproblem}[]{Open Problem}
%\usepackage{ulem}

\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\numberwithin{equation}{section}
\numberwithin{figure}{section}
% references alphabetically
% make relation clearer
\def\chk{\marginpar{\quad\qquad\textbf{check!}}}

\def\d{\textrm{d}}

\def\des{\mathsf{des}}
\def\ides{\mathsf{ides}}
\def\iasc{\mathsf{iasc}}
\newcommand{\asc}{\mathsf{asc}}
\def\max{\mathsf{max}}
\def\zero{\mathsf{zero}}
\def\rep{\mathsf{rep}}
\def\rmin{\mathsf{rmin}}
\def\lmax{\mathsf{lmax}}
\def\lmin{\mathsf{lmin}}
\def\rmax{\mathsf{rmax}}
\def\nasc{\mathsf{nasc}}
\def\sebr{\mathsf{sebr}}
\def\Prm{\mathsf{Prm}}
\def\Prmin{\mathsf{Prmin}}
\def\Rmin{\mathsf{Rmin}}
\def\mmasc{\mathsf{mmasc}}


\def\ealm{\mathsf{ealm}}
\def\zpair{\mathsf{zpair}}
\def\mpair{\mathsf{mpair}}
\def\zpos{\mathsf{zpos}}
\def\mpos{\mathsf{mpos}}
\def\rpos{\mathsf{rpos}}

\def\A{\mathcal{A}}
\def\T{\CMcal{T}}
\def\M{\CMcal{M}}
\def\R{\CMcal{R}}
\def\I{\operatorname{{\bf I}}}
\def\CS{ \CMcal{S}}
\def\B{ \mathcal{B}}
\def\C{ \CMcal{C}}
\def\D{ \CMcal{D}}
\def\G{ \CMcal{G}}
\def\Gf{ \mathfrak{G}}

%set-valued statistics
\def\IDES{\mathrm{IDES}}
\def\IASC{\mathrm{IASC}}
\def\DES{\mathrm{DES}}
\def\RMAX{\mathrm{RMAX}}
\def\LMAX{\mathrm{LMAX}}
\def\LMIN{\mathrm{LMIN}}
\def\DIST{\mathrm{DIST}}
\def\ASC{\mathrm{ASC}}
\def\RMIN{\mathrm{RMIN}}
\def\ZERO{\mathrm{ZERO}}
\def\MAX{\mathrm{MAX}}
\def\NASC{\mathrm{NASC}}
\def\RUN{\mathrm{RUN}}
\def\ST{\mathrm{ST}}

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

\def\boxit#1{\leavevmode\hbox{\vrule\vtop{\vbox{\kern.33333pt\hrule\kern1pt\hbox{\kern1pt\vbox{#1}\kern1pt}}\kern1pt\hrule}\vrule}}

\usepackage{collectbox}
\makeatletter
\newcommand{\mybox}{%
	\collectbox{%
		\setlength{\fboxsep}{1pt}%
		\fbox{\BOXCONTENT}%
	}%
}
\makeatother
%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\PATTERN}{
	\draw[step=1, xshift=14pt, yshift=14pt, \cfill, line cap=round] (0,0) grid (3,3);
	\draw[step=1, xshift=14pt, yshift=14pt, thick] (0,2) -- (3,2);
	\draw[step=1, xshift=14pt, yshift=14pt, thick] (1,0) -- (1,3);
	\foreach \x/\y in {1/1,2/3,3/2} \node[disc, fill=black] at (\x,\y) {};
}

\newcommand{\SEPATTERN}{
	\draw[step=1, xshift=14pt, yshift=14pt, \cfill, line cap=round] (0,0) grid (3,3);
	\draw[step=1, xshift=14pt, yshift=14pt, thick] (0,1) -- (3,1);
	\draw[step=1, xshift=14pt, yshift=14pt, thick] (1,0) -- (1,3);
	\foreach \x/\y in {1/2,2/3,3/1} \node[disc, fill=black] at (\x,\y) {};
}

\newcommand{\pattern}{\!\raisebox{-0.3em}{
		\begin{tikzpicture}[line width=0.7pt, scale=0.15]
		\tikzstyle{disc} = [circle,thin,draw=black, minimum size=1.7pt, inner sep=0pt ]
		\PATTERN
		\end{tikzpicture}}
}

\newcommand{\Pattern}{\raisebox{-0.3em}{
		\begin{tikzpicture}[line width=0.7pt, scale=0.3]
		\tikzstyle{disc} = [circle,thin,draw=black, minimum size=4pt, inner sep=0pt ]
		\PATTERN
		\end{tikzpicture}}
}

\newcommand{\sepattern}{\!\raisebox{-0.3em}{
		\begin{tikzpicture}[line width=0.7pt, scale=0.15]
		\tikzstyle{disc} = [circle,thin,draw=black, minimum size=1.7pt, inner sep=0pt ]
		\SEPATTERN
		\end{tikzpicture}}
}

\newcommand{\Sepattern}{\raisebox{-0.3em}{
		\begin{tikzpicture}[line width=0.9pt, scale=0.3]
		\tikzstyle{disc} = [circle,thin,draw=black, minimum size=4pt, inner sep=0pt ]
		\SEPATTERN
		\end{tikzpicture}}
}

\newcommand{\subi}{\textsf{Rem1}}
\newcommand{\subii}{\textsf{Rem2}}
\newcommand{\subiii}{\textsf{Rem3}}

\newcommand{\addi}{\textsf{Add1}}
\newcommand{\addii}{\textsf{Add2}}
\newcommand{\addiii}{\textsf{Add3}}

\newcommand{\cfill}{black!40}
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\newcommand{\nodestyle}{\tikzstyle{every node} = [font=\footnotesize]}
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	[ circle,thin,fill=\cfilll,draw=black, minimum size=\ns, inner sep=0pt ] }
\newcommand{\style}{
	\nodestyle
	\discstyle
}
%% define your title in the usual way
\title[Bi-symmetric multiple equidistributions on ascent sequences]{Bi-symmetric multiple equidistributions\\ on ascent sequences}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details

\author[Emma Yu Jin and Michael J. Schlosser]{Emma Yu Jin\thanks{\href{mailto:yu.jin@univie.ac.at}{yu.jin@univie.ac.at}.
Partially supported by Austrian Science Fund FWF, project P 32305.}\addressmark{1}
and  Michael J. Schlosser\thanks{\href{mailto:michael.schlosser@univie.ac.at}{michael.schlosser@univie.ac.at}.
Partially supported by Austrian Science Fund FWF, project P 32305.}\addressmark{1}}
%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Fakult\"{a}t f\"{u}r Mathematik,
Universit\"{a}t Wien, Vienna, Austria}

%% put the date of submission here
\received{December 1, 2020}

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%\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 since the seminal work by Bousquet-M\'elou, Claesson,
  Dukes and Kitaev (2010) that certain refinements of the ascent sequences
  with respect to several natural statistics are in bijection with
  corresponding refinements of $({\bf2+2})$-free posets and permutations
  that avoid a bivincular pattern. Different multiply-refined enumerations
  of ascent sequences and other bijectively equivalent structures have
  subsequently been extensively studied by various authors.
  
  We contribute new bi-symmetric equidistributions to this subject. Our main result is
  a bijective proof of a bi-symmetric septuple equidistribution of
  statistics on ascent sequences, involving
  the number of ascents ($\mathsf{asc}$),
  the number of repeated entries ($\mathsf{rep}$),
  the number of zeros ($\mathsf{zero}$),
  the number of maximal entries ($\mathsf{max}$),
  the number of right-to-left minima ($\mathsf{rmin}$),
  and two additional statistics.
  We further establish a new transformation formula for non-terminating
  basic hypergeometric $_4\phi_3$ series expanded as an analytic function
  in base $q$ around $q=1$, which is utilized to prove two (bi)-symmetric
  quadruple equidistributions on ascent sequences. 
  
  A by-product of our findings includes the affirmation of a conjecture about the bi-symmetric equidistribution between the quadruples of Euler--Stirling statistics
  $(\mathsf{asc},\mathsf{rep},\mathsf{zero},\mathsf{max})$ and
  $(\mathsf{rep},\mathsf{asc},\mathsf{max},\mathsf{zero})$ on ascent
  sequences, that was motivated by a double Eulerian equidistribution due
  to Foata (1977) and recently proposed by Fu, Lin, Yan, Zhou and the
  first author (2018).}

%% put your French abstract here, or comment this out if you don't have one
%\resume{\lipsum[2]}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{ascent sequences, equidistributions, Euler--Stirling statistics,
Fishburn numbers, basic hypergeometric series}

%% 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
\usepackage[backend=bibtex]{biblatex}
\addbibresource{64Jin.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}

In the seminal paper \cite{MR2652101} by Bousquet-M\'elou, Claesson, Dukes
and Kitaev, ascent sequences were introduced, as they are in bijection with
several different combinatorial structures such as $({\bf2+2})$-free posets,
certain bivincular pattern-avoiding permutations, Stoimenow's involution
and regular linearized chord diagrams~\cite{MR1611987,MR1860536}.
Several natural statistics on posets, permutations and sequences are also
kept track of by a sequence of bijections established by these authors.
Since then, various joint distributions of classical statistics on ascent
sequences and many other bijectively equivalent structures including
Fishburn matrices~\cite{MR253942,MR776781} and $({\bf 2-1})$-avoiding
inversion sequences have been intensively explored; see for instance
\cite{MR2736327,MR2607339,MR2847912,MR2871752,MR3388862,MR2832334,MR3668878,MR2971707}.

Recently, Fu, Lin, Yan, Zhou and the first author~\cite{MR4011082}
discovered a new decomposition of ascent sequences which contributes
to a systematic study of Eulerian and Stirling statistics on ascent
sequences, certain pattern-avoiding permutations and
$({\bf 2-1})$-avoiding inversion sequences. In particular, their work
led them to conjecture the bi-symmetry of a quadruple Euler--Stirling
statistics on ascent sequences (see Conjecture~\ref{conj1ref}) that is
motivated by a double Eulerian equidistribution due to Foata~\cite{MR519777}.
However, it appears that the use of the new decomposition from \cite{MR4011082}
is not sufficient to prove this bi-symmetry conjecture.

In our present work, we affirm this conjecture in two different ways:
one by developing a second new decomposition of ascent sequences; and the
other one by identifying the generating function of the quadruple statistics
as a basic hypergeometric series to which a (newly obtained)
transformation formula is applied. We start with some necessary
definitions and then state the consequences of our results.

\smallskip
An {\em inversion sequence} $(s_1,s_2,\ldots,s_n)$ is a sequence of
non-negative integers such that for all $i$, $0\le s_i<i$. We denote
by $\CMcal{I}_n$ the set of inversion sequences of length $n$, which is
in one-to-one correspondence with the set $\mathfrak{S}_n$ of permutations
of $[n]:=\{1,2,\ldots,n\}$ via the well known Lehmer code $\sigma$
(see for instance \cite{MR519777,MR0113289}).
That is, for $\pi=\pi_1\pi_2\cdots \pi_n\in \mathfrak{S}_n$, the map
$\sigma:\mathfrak{S}_n\rightarrow \CMcal{I}_n$ is defined as
\begin{align*}
\sigma(\pi)=(s_1,s_2,\ldots,s_n),
\quad\mbox{ where } s_i:=|\{j:j<i \,\mbox{ and }\, \pi_j>\pi_i\}|.
\end{align*}
Some restrictions set up on permutations and inversion sequences could produce
new sets of equal cardinality, but not necessarily through the Lehmer code.
For instance,
%As a case in point, two kinds, respectively, of restricted inversion
%sequences and of restricted permutations, known as 
ascent sequences and
$(\sepattern\,)$-avoiding permutations (defined as below) are equinumerous.
\begin{definition}[Ascent sequence]
	For any sequence $s\in\CMcal{I}_n$, let
	\begin{align}\label{E:asc}
	\asc(s)&:=|\{i\in[n-1]: s_i<s_{i+1}\}|
	\end{align}
	be the number of {\bf asc}ents of $s$.
	An inversion sequence $s\in \CMcal{I}_n$ is an {\em ascent sequence}
	if for all $2\leq i\leq n$, the $s_i$ satisfy
	\begin{align*}
	s_i\leq \asc(s_1,s_2,\ldots, s_{i-1})+1.
	\end{align*} 
\end{definition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{definition}[$(\sepattern\,)$-avoiding permutation]
We say that a permutation $\pi\in\mathfrak{S}_n$ {\em avoids the pattern
\mbox{\sepattern}} if there is no subsequence $\pi_i\pi_{i+1}\pi_j$  of $\pi$
satisfying both $\pi_i-1=\pi_j $ and $\pi_i<\pi_{i+1}$. Otherwise we say $\pi$
contains the pattern \mbox{\sepattern}. Sometimes the pattern \mbox{\sepattern}
is written as $2|3\bar{1}$.
\end{definition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The $(\sepattern\,)$-avoiding permutations, more generally, permutations
that avoid a specific bivincular pattern, 
were introduced and studied by
Bousquet-M\'elou, Claesson, Dukes and Kitaev~\cite{MR2652101} as both of
them are surprisingly in bijection with other classical combinatorial
structures such as $({\bf2+2})$-free posets~\cite{MR253942,MR776781} and
regular linearized chord diagrams~\cite{MR1611987,MR1860536}.

Let $\A_n$ and $\mathfrak{S}_n(\sepattern)$ be the sets respectively of ascent
sequences and $(\sepattern\,)$-avoiding permutations of length $n$.
Bousquet-M\'elou, Claesson, Dukes and Kitaev~\cite{MR2652101} proved that
\begin{equation}\label{gffb}
\left|\A_n\right|=\left|\mathfrak{S}_n(\sepattern)\right|=
[t^n]\sum_{k=1}^{\infty}\prod_{i=1}^k(1-(1-t)^i),
\end{equation}
and thus, as a consequence of a result by Zagier~\cite{MR1860536}
(who discovered that the series on the right-hand side of \eqref{gffb} is
the generating functions of the Fishburn numbers), $\left|\A_n\right|$
is equal to the $n$-th Fishburn number (see A022493 of the OEIS~\cite{OEIS}).
Their explicit values are given as
\begin{equation*}
  \left(\left|\A_n\right|\right)_{n\ge 1}=
  (1, 2, 5, 15, 53, 217, 1014, 5335, 31240, 201608, \ldots),
\end{equation*}
for which no closed form is known.
The study of Fishburn numbers and their generalizations has remarkably led
to many interesting results, including congruences \cite{MR3435730,MR3345301},
asymptotic formulas~\cite{MR3219259,HwangJin,MR1860536}, intriguing
connections to transformations of hypergeometric series~\cite{MR3168523},
modular forms~\cite{MR3219259,MR1860536} and a variety of
bijections~\cite{MR2736327,MR2607339,MR2847912,MR2871752,MR3388862,MR2832334,MR3668878,MR2971707}.
In particular, various members of the Fishburn family can be viewed as supersets of corresponding members of the Catalan family. Here the Fishburn (resp. Catalan) family refers to classes of combinatorial objects enumerated by the Fishburn (resp. Catalan) numbers.
%(which are defined
%to be classes of combinatorial objects enumerated by the Fishburn numbers)
%can be viewed as supersets of corresponding members of the Catalan family 
%(i.e., classes of combinatorial objects enumerated by the Catalan numbers,
%such as non-crossing matchings, $231$-avoiding permutations and Dyck paths). 

This extended abstract, which is essentially an abridged version of
\cite{JinSchlosser}, is devoted to the presentation of new bijective
and basic hypergeometric aspects of Fishburn structures.

Let us review some classical statistics on ascent sequences and
$(\sepattern\,)$-avoiding permutations. For any sequence $s\in\CMcal{I}_n$, 
$\asc(s)$ is defined in \eqref{E:asc}. Let furthermore
\begin{align*}
\rep(s)&:=n-\vert\{s_1,s_2,\ldots,s_n\}\vert,\\
\zero(s)&:=\vert\{i\in[n]: s_i=0\}\vert,\\
\max(s)&:=\vert\{i\in[n]:s_i=i-1\}\vert,\quad\text{and}\\
\rmin(s)&:=|\{s_i: s_i<s_j\text{ for all $j>i$}\}|,
\end{align*}
be the respective numbers of {\bf rep}eated entries,  {\bf zero}s,
{\bf maxi}mal entries (or maximals for short) and {\bf r}ight-to-left
{\bf min}ima of $s$. For instance, when $s=(0,1,2,0,1,3,5)\in \CMcal{I}_7$,
then $\asc(s)=5$, $\rep(s)=2$, $\zero(s)=2$, $\max(s)=3$ and $\rmin(s)=4$.

For any permutation $\pi\in\mathfrak{S}_n$, let
\begin{align*}
\des(\pi)&:=|\{i\in[n-1]: \pi_i>\pi_{i+1}\}|,\\
  \iasc(\pi)&:=\asc(\pi^{-1})=|\{i\in[n-1]:\pi_i+1
              \mbox{ appears to the right of }\pi_i\}|,
\end{align*}
be the number of {\bf des}ents and {\bf i}nverse {\bf asc}ents of $\pi$,
respectively. Similar to $\rmin$, the statistics $\lmin$, $\lmax$ and $\rmax$
represent the numbers of {\bf l}eft-to-right {\bf min}ima, {\bf l}eft-to-right
{\bf max}ima and  {\bf r}ight-to-left {\bf max}ima, respectively.

Previous bijections developed in \cite{MR2652101,MR2607339,MR4011082}
preserve natural statistics on posets, permutations, sequences and matrices.
As examples, we list below five pairs of equidistributed statistics that were
established in those papers.
\begin{align*}
(\asc,\zero) \mbox{ on ascent sequences}
&\xleftrightarrow{1-1} (\des,\lmax)
\mbox{ on (\sepattern)-avoiding permutations},\\
&\xleftrightarrow{1-1}(\mathsf{mag},\mathsf{min})
\mbox{ on } ({\bf2+2})\mbox{-free posets},\\
&\xleftrightarrow{1-1} (\mathsf{dim}, \mathsf{rowsum}_1)
\mbox{ on Fishburn matrices},\\
&\xleftrightarrow{1-1} (\rep,\max) \mbox{ on } ({\bf2-1})
\mbox{-avoiding inversion sequences}.
\end{align*}
\begin{remark}
The statistics $\mathsf{mag}$, $\mathsf{min}$ are abbreviations for
{\bf mag}nitude and the number of {\bf min}imal elements of a poset; the
statistics $\mathsf{dim}$ and  $\mathsf{rowsum}_1$ refer to {\bf dim}ension
and the {\bf sum} of entries in the {\bf first row} of a matrix. 
\end{remark}
%For convenience, all aforementioned statistics are classified into two categories as in \cite{MR4011082}: any statistic whose distribution over a member of Fishburn family equals the distribution of $\asc$ (resp. $\zero$) on ascent sequences is called {\em an Eulerian} (resp. a Stirling) statistic. 

In a recent paper \cite{MR4011082} by Fu, Lin, Yan, Zhou and the
first author, a joint {\em symmetric} distribution of statistics
$\asc$ and $\rep$ over ascent sequences was discovered. The motivation
came from a symmetric distribution of $(\asc,\rep)$ on inversion sequences
\begin{align}\label{inv:sym}
\sum_{s\in\CMcal{I}_n}
u^{\asc(s)}x^{\rep(s)}
&=\sum_{s\in\CMcal{I}_n}
u^{\rep(s)}x^{\asc(s)}.
\end{align}
This is a direct consequence of a double Eulerian equidistribution due to
Foata~\cite{MR519777}:
\begin{equation}\label{dou:fo}
\sum_{s\in\CMcal{I}_n}
u^{\asc(s)}x^{\rep(s)}=\sum_{\pi\in\mathfrak{S}_n}u^{\des(\pi)}x^{\iasc(\pi)}.
\end{equation}
It turns out that not only \eqref{inv:sym} and \eqref{dou:fo} are true if
$\CMcal{I}_n$ and $\mathfrak{S}_n$ are replaced by the corresponding
subsets $\A_n$ and $\mathfrak{S}_n(\sepattern)$, but an even stronger result
on a bi-symmetric equidistribution of Euler--Stirling
statistics\footnote{We adopt the classification of statistics from
  \cite{MR4011082}: any statistic whose distribution over a member of
  the Fishburn family equals the distribution of $\asc$ (resp.\ $\zero$)
  on ascent sequences is called {\em an Eulerian} (resp.\ a Stirling)
  statistic. So according to Theorem~\ref{bij:sym}, $\asc,\rep$
 are Eulerian statistics and $\zero,\max,\rmin$ are Stirling statistics.} 
over ascent sequences was conjectured. 
%appears to hold, supported by experimental data.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{conjecture}[\cite{MR4011082}]\label{conj1ref}
For each $n\geq1$, the following bi-symmetric quadruple equidistribution holds:
	\begin{align*}
	\sum_{s\in\A_n}
	u^{\asc(s)}x^{\rep(s)}z^{\zero(s)}y^{\max(s)}
	&=\sum_{s\in\A_n}
	u^{\rep(s)}x^{\asc(s)}z^{\max(s)}y^{\zero(s)}. %\mbox{ or equivalently }\\
	%	\sum_{s\in\mathfrak{S}_n(\sepattern)}
	%u^{\des(s)}x^{\iasc(s)}z^{\lmax(s)}q^{\lmin(s)}
	%&=\sum_{s\in\mathfrak{S}_n(\small{\sepattern})}
	%u^{\iasc(s)}x^{\des(s)}z^{\lmin(s)}q^{\lmax(s)}.
	\end{align*}
\end{conjecture}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{remark}
  Conjecture~\ref{conj1ref} is equivalent to a bi-symmetric equidistribution
  between the quadruples $(\des,\iasc,\lmax,\lmin)$ and
  $(\iasc,\des,\lmin,\lmax)$ on $(\sepattern)$-avoiding permutations,
  according to \cite [Theorem~12]{MR4011082}.
\end{remark}
Two results in approaching this conjecture were presented in
\cite{MR4011082}: one is an explicit formula for the generating function
of ascent sequences with respect to the four statistics
$\asc,\rep,\zero,\max$ (see Theorem~\ref{T:gen}); and the other one is
a quadruple equidistribution between $(\asc,\rep,\zero,\max)$ and
$(\rep,\asc,\rmin,\zero)$ on ascent sequences (see Theorem~\ref{bij:sym}).

Let $\G(t;x,y,u,z)$ denote the generating function of ascent sequences
counted by the length (variable $t$), $\asc$ (variable $u$), $\rep$
(variable $x$), $\max$ (variable $y$) and $\zero$ (variable $z$). That is,
\begin{align}\label{E:g1}
\G(t;x,y,u,z):=\sum_{n=1}^{\infty}t^n\sum_{s\in\A_n}
x^{\rep(s)}y^{\max(s)}u^{\asc(s)}z^{\zero(s)}.
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}[\cite{MR4011082}]\label{T:gen}
The generating function $\G(t;x,y,u,z)$ of ascent sequences is
\begin{align}\label{E:genG}
\G(t;x,y,u,z)=\sum_{m=0}^{\infty}&\frac{zyr x^m(1-yr)(1-r)^m(x+u-xu)}
{[x(1-u)+u(1-yr)(1-r)^m][x+u(1-x)(1-yr)(1-r)^m]}\notag\\
&\times \prod_{i=0}^{m-1}\frac{1+(zr-1)(1-yr)(1-r)^i}{x+u(1-x)(1-yr)(1-r)^i},
\end{align}
where $r=t\,(x+u-xu)$.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}[\cite{MR4011082}]\label{bij:sym}
  There is a bijection $\Upsilon:\A_n\rightarrow\A_n$ which transforms
  the quadruple
$$
(\asc,\rep,\zero,\max)\; \text{ to }\; (\rep,\asc,\rmin,\zero).
$$
%Consequently, $(\asc,\rep)$ is symmetric on $\A_n$, $\rep$ is a Eulerian statistic and $\max,\rmin$ are Stirling statistics.
\end{theorem}
Conjecture~\ref{conj1ref} can be settled, with the the help of
Theorems~\ref{T:gen} and \ref{bij:sym}, by showing either (I) or (II),
described as follows.

\begin{enumerate}[label=(\Roman*)]
\item $\G(t;x,y,u,z)=\G(t;u,z,x,y)$;
\item the quadruple $(\asc,\rep,\zero,\max)$ has the same
  distribution as $(\asc,\rep,\zero,\rmin)$ over ascent sequences. 
\end{enumerate}
We are able to settle Conjecture~\ref{conj1ref} independently
in both ways, (I) and (II).

\section{Main results}
\subsection{A bi-symmetric septuple equidistribution}
Our first main result (Theorem~\ref{T:5tuple}) is a bijective proof of
a bi-symmetric \textit{septuple} equidistribution on ascent sequences,
which significantly generalizes~(II) and consequently affirms
Conjecture~\ref{conj1ref}.

The five statistics $\asc$, $\rep$, $\zero$, $\max$, and $\rmin$
on inversion sequences were defined in the previous section.
We now define two additional statistics on ascent sequences.

First we recall the definition of the statistic $\ealm$ on ascent sequences
which was introduced in \cite{MR4011082}.
Let $s$ be an ascent sequence with $\mathsf{max}(s)\ne |s|$,
then $\ealm(s)=s_{\max(s)+1}$, i.e., the entry right after the last maximal entry.
For the ascent sequence $s=(0,1,\ldots,|s|-1)$ that has $\max(s)=|s|$, we set $\ealm(s)=0$.
	
	For example, $\ealm(0,1,{\bf 0},1,3,0,2)=0$.

Next, we introduce the new statistic $\rpos$ on ascent sequences.
For the sake of convenience, we index all right-to-left minima from left to right starting from 0 (rather than from 1). For any ascent sequence $s$ with $\rmin(s)\ne |s|$, define
$\rpos(s)=m$ if $m$ is the maximal index such that the $m$-th
right-to-left minimum appears at least twice after the $(m-1)$-th
right-to-left minimum. If no such $m$ exists or $\rmin(s)=|s|$, then
set $\rpos(s)=0$.
	
For example, $\rpos({\bf 0},{\bf 0},1,2,3,4)=0$ and
$\rpos(0,0,1,2,0,1,2,1,{\bf 3},{\bf 3},4)=2$.

Having defined the two additional statistics $\ealm$ and $\rpos$,
we are now ready to state our main result.
\begin{theorem}\label{T:5tuple}
  There is a bijection $\Phi: \A_n\rightarrow \A_n$ such that
  for all $s\in \A_n$, 
\begin{align}\label{E:thm7sta}
  (\asc,\rep,\zero,\max,\ealm,\rmin,\rpos)s=
  (\asc,\rep,\zero,\rmin,\rpos,\max,\ealm)\Phi(s).
\end{align}
\end{theorem}
%Inspired by this new statistic, a new decomposition of ascent sequences is discovered, which provides a crucial piece of the puzzle solved here.
The main idea to prove Theorem~\ref{T:5tuple} relies on two
{\em parallel} decompositions of ascent sequences that are in close
relation to the two respective statistics $\ealm$ and $\rpos$.
The first decomposition was discovered in \cite{MR4011082}, while the
second decomposition is new and plays an essential role in the proof
of Theorem \ref{T:5tuple}.
The proof itself consists of a sequence of bijections 
with delicate subdivisions into cases. See \cite{JinSchlosser} for the details.

%in parallel with the one from \cite{MR4011082}, plays a crucial role in solving the puzzle here. 

%The construction of $\Phi$ in Theorem \ref{T:5tuple} is built upon a new decomposition of ascent sequences inspired by the statistic $\rpos$, which plays a crucial role in solving the puzzle here. 

\subsection{A new basic hypergeometric transformation
  with applications}
\begin{sloppypar}
Our second main result (Theorem~\ref{thm:4phi3tf}) is a new transformation
formula of non-terminating basic hypergeometric ${}_4\phi_3$ series,
valid as an identity expanded in base $q=1-r$ around $q=1$, or,
equivalently, $r=0$. Basic hypergeometric series in base $q$
expanded around $q=1$ typically appear as (multiply-refined)
generating functions of objects of the Fishburn family
(compare with \eqref{gffb}) which is the reason of our interest
in identities for such series, and, indeed, we successfully
apply special cases of the new ${}_4\phi_3$ transformation to
prove equidistribution results.
\end{sloppypar}

For convenience, we recall some standard notions from the theory of basic
hypergeometric series, cf.\ \cite{MR2128719}.

For indeterminates $a$ and $q$ (the latter is referred to as the base),
and non-negative integer $k$,
the basic shifted factorial (or $q$-shifted factorial) is defined as
\begin{equation*}
(a;q)_k:=\prod_{j=1}^k(1-aq^{j-1}).
\end{equation*}
This also makes sense for $k=\infty$, where the infinite product is viewed as
a formal power series in $q$ (whereas, viewed as an analytic expression in $q$,
we would need to insist on $|q|<1$, for convergence).
When dealing with products of $q$-shifted factorials, it is convenient to use
the following short notation,
\begin{equation*}
(a_1,\ldots,a_m;q)_k:=(a_1;q)_k\cdots(a_m;q)_k,
\end{equation*}
where again $k$ is a non-negative integer or $\infty$.

An ${}_\alpha\phi_\beta$ basic hypergeometric series with $\alpha$
upper parameters $a_1,\ldots,a_\alpha$, and $\beta$ lower parameters
$b_1,\ldots,b_\beta$, base $q$ and argument $z$ is defined as
\begin{equation}\label{eq:bhyp}
{}_\alpha\phi_\beta\!\left[\begin{matrix}a_1,\ldots,a_\alpha\\
b_1,\ldots,b_\beta\end{matrix};q,z\right]:=
\sum_{k=0}^\infty\frac{(a_1,\ldots,a_\alpha;q)_k}
{(q,b_1,\ldots,b_\beta;q)_k}\left((-1)^kq^{\binom k2}\right)^{1+\beta-\alpha}z^k.
\end{equation}
The series in \eqref{eq:bhyp} (where the lower parameters are assumed
to be chosen such that no poles occur in the summands of the series)
terminates if one of the upper parameters, say $a_1$,
is of the form $q^{-n}$. Since $(q^{-n};q)_k=0$ for $k>n$,
the series in that case contains only finitely many non-vanishing terms.
If the series does not terminate, one usually imposes $|q|<1$.
See \cite[Sec.~1.2]{MR2128719} for conditions under which the series converges.

One of the most important identities in the theory of basic
hypergeometric series is the Sears transformation~\cite[(III.15)]{MR2128719},
\begin{equation}\label{tfsears}
{}_4\phi_3\!\left[\begin{matrix}q^{-n},a,b,c\\
d,e,abcq^{1-n}/de\end{matrix};q,q\right]=
\frac{(e/a,de/bc;q)_n}{(e,de/abc;q)_n}
\,{}_4\phi_3\!\left[\begin{matrix}q^{-n},a,d/b,d/c\\
d,aq^{1-n}/e,de/bc\end{matrix};q,q\right].
\end{equation}
In \eqref{tfsears}, $a,b,c,d,e$ and $q$ are indeterminates
and $n$ is a non-negative integer (which is responsible
that both ${}_4\phi_3$ series are actually finite sums and each
contains only $n+1$ non-vanishing terms).

With the relevant definitions and ingredients of proof given,
we are ready to state the main result of this subsection.
\begin{theorem}\label{thm:4phi3tf}
Let $a,b,c,d,e,r$ be complex variables, $j$ be a non-negative integer.
Then, assuming that none of the denominator factors in \eqref{tf43}
have vanishing constant term in $r$, we have the following transformation
of convergent power series in $a$ and $r$:
\begin{align}\label{tf43}
&{}_4\phi_3\!\left[\begin{matrix}(1-r)^j,1-a,b,c\\
d,e,(1-r)^{j+1}(1-a)bc/de\end{matrix};1-r,1-r\right]\notag\\
&=\frac{((1-r)/e,(1-r)(1-a)bc/de;1-r)_j}{((1-r)(1-a)/e,(1-r)bc/de;1-r)_j}
\notag\\
&\quad\;\times{}_4\phi_3\!\left[\begin{matrix}(1-r)^j,1-a,d/b,d/c\\
d,de/bc,(1-r)^{j+1}(1-a)/e\end{matrix};1-r,1-r\right].
\end{align}
\end{theorem}

While for non-terminating basic hypergeometric series in base $q$
we usually consider expansions around $q=0$, here
(and more generally, when dealing with generating functions
of members of the Fishburn family)
we are dealing with power series in $r$, which
can be written as basic hypergeometric series in base $q=1-r$,
thus can be viewed as functions analytic around $q=1$.
We need to be cautious when we resort to non-terminating
identities for basic hypergeometric series.

The first part of the argument in the
proof of Theorem \ref{thm:4phi3tf}, as our main result in this section,
is similar to that used by Andrews and Jel\'inek in \cite{MR3168523}
for establishing $q$-series identities around $q=1$.

\begin{proof}[Proof of Theorem \ref{thm:4phi3tf}]
Observe that both sides of
the identity converge as power series in $a$, thus
are analytic functions in $a$.
%(This would not be true if $r$ would be replaced by $1-r$
%with the series being expanded around $r=0$.)
Indeed, for each $m\ge 0$ the expansion of
$(1-a;1-r)_m$ in monomials $a^ir^l$ only involves terms with $i+l\ge m$
and each factor in the denominator of the series
has a non-vanishing constant term. Thus, in the expansion of the
series in the variables $a$ and $r$ the contribution of
coefficients for each monomial $a^ir^l$ is finite.

Now both sides of \eqref{tf43} agree for $a=1-(1-r)^{-n}$ where
$n=0,1,2,\dots$ by the $(q,a,b,c,d,e)\mapsto
(1-r,(1-r)^j,b,c,d,e)$ special case
of the transformation in \eqref{tfsears}.
Since we have shown \eqref{tf43}
for infinitely many values of
$a$ accumulating at $a=-\infty$, i.e. $1-a=\infty$,
%(the transformation \eqref{tfsears}
%itself is valid in the limiting case $n\to\infty$ (i.e. $q^{-n}\to\infty$)!),
by the identity theorem in complex analysis
the transformation \eqref{tf43}
is true for all $a$ in its domain of analyticity.
\end{proof}

We utilize special cases of Theorem~\ref{thm:4phi3tf} to give
analytic proofs of two different quadruple (bi)-symmetric equidistributions
of Euler--Stirling statistics on ascent sequences, collected in
Theorem~\ref{TC:int}.
The first application of Theorem~\ref{thm:4phi3tf} is a proof of (I)
by making use of the explicit form of the generating function in
Theorem~\ref{T:gen}, and thus constitutes a non-combinatorial proof
of the bi-symmetric equidistribution in Conjecture~\ref{conj1ref},
while the second application establishes a symmetric equidistribution
by employing a new explicit generating function obtained by
a refined recursive construction of ascent sequences from~\cite{MR4011082}.
\begin{theorem}\label{TC:int}
For the generating function defined in \eqref{E:g1}, we have the bi-symmetry
\begin{equation}\label{E:sym1}
\G(t;x,y,u,z)=\G(t;u,z,x,y).
\end{equation}
Furthermore, define 
	\begin{equation}\label{E:g2}
	\Gf(t;x,y,u,v):=\sum_{n=1}^{\infty}t^n\sum_{s\in\A_n}x^{\rep(s)}y^{\max(s)}
	u^{\asc(s)}v^{\rmin(s)},
	\end{equation}
	then we have, with $r=t(x+u-xu)$,
	\begin{align}\label{E:g5}
	\Gf(t;x,y,u,v)=\frac{vyt}{1-vytu}
	+\sum_{m=0}^{\infty}\frac{rv(1-yr)(1-r)^m}
	{(x-xu+u(1-yr)(1-r)^m)(1-tuvy)}&\notag\\
	\times\prod_{i=0}^{m}\frac{x(1-(1-yr)(1-r)^i)(x-xu+u(1-yr)(1-r)^i)}
	{(x-u(x-1)(1-yr)(1-r)^i)(x-xu+u(1-rv)(1-yr)(1-r)^i)}&,
	\end{align}
and the symmetry
	\begin{equation}\label{E:sym2}
	\Gf(t;x,y,u,v)=\Gf(t;x,v,u,y),
	\end{equation}
\end{theorem}
\begin{remark}
In the language of bijections, the (bi)-symmetric equidistributions in Theorem \ref{TC:int} mean that for any ascent sequence $s\in\A_n$, 
\begin{align*}
(\asc,\rep,\zero,\max)s&=(\rep,\asc,\max,\zero)\Upsilon^{-1}(\Phi(s)),\\
(\asc,\rep,\max,\rmin)s&=(\asc,\rep,\rmin,\max)\Phi(s),\\
(\asc,\rep,\zero,\rmin)s&=(\rep,\asc,\rmin,\zero)\Upsilon(\Phi(s)),
\end{align*}
where $\Upsilon$ and $\Phi$ are the bijections respectively in
Theorems~\ref{bij:sym} and \ref{T:5tuple}. 
\end{remark}

\begin{remark}
We are not the first ones to study equivalent forms for generating functions of
objects of the Fishburn family using tools from basic hypergeometric series.
Initiating with work of Zagier~\cite{MR1860536} who established the basic
hypergeometric series in \eqref{gffb} as a concrete form of the generating
function $\G(t;1,1,1,1)$ for the Fishburn numbers, Andrews and
Jel\'{i}nek~\cite{MR3168523} subsequently proved three equivalent forms of
$\G(t;1,1,1,z)$ by applying the Rogers--Fine identity. However, to the best of our
knowledge, no algebraic or analytic arguments to determine equivalent forms of
the generating functions $\G(t;x,y,u,z)$ or $\Gf(t;x,y,u,v)$ were known, not even,
say, for the special case $\G(t;1,1,u,z)$. Our analytic proofs of
$\G(t;x,y,u,z)=\G(t;u,z,x,y)$ and $\Gf(t;x,y,u,v)=\Gf(t;x,v,u,y)$ strengthen the
already known existing ties between (refined) generating functions of objects of
the Fishburn family with basic hypergeometric series that are expanded in base
$q=1-r$ around $r=0$. At the same time it demonstrates the benefit of having
equivalent forms of generating functions, and the power of basic hypergeometric machinery.
\end{remark}

\section{Discussion}
\subsection{Reformulations}
All aforementioned (bi)-symmetric distributions on ascent sequences have
counterparts over other members of the Fishburn family.
\begin{corollary}
There are three bijections between $\mathfrak{S}_n(\sepattern)$ and
itself such that the following three (bi)-symmetric equidistributions
hold, respectively:
\begin{align*}
  (\des,\iasc,\lmax,\lmin,\rmax)\pi
  &=(\des,\iasc,\lmax,\rmax,\lmin)(\Psi^{-1}\circ\Phi\circ\Psi)(\pi),\\
  (\des,\iasc,\lmax,\lmin)\pi
  &=(\iasc,\des,\lmin,\lmax)(\Psi^{-1}\circ\Upsilon^{-1}
    \circ\Phi\circ\Psi)(\pi),\\
  (\des,\iasc,\lmax,\rmax)\pi
  &=(\iasc,\des,\rmax,\lmax)(\Psi^{-1}\circ\Upsilon\circ\Phi\circ\Psi)(\pi),
\end{align*}
where $\Upsilon,\Phi$ are the bijections respectively in
Theorems~\ref{bij:sym} and \ref{T:5tuple}, and
$\Psi:\mathfrak{S}_n(\sepattern)\rightarrow \A_n$ is the bijection
from \cite[Theorem~12]{MR4011082}.
\end{corollary}
Let us recall the definition of Fishburn matrices and associated
three Stirling statistics.

Any cell $(i,j)$ of a matrix $M$ is called a
\emph{weakly north-east cell} if $M_{i,j}\ne 0$ and $M_{s,t}=0$
for all $s\le i$ and $t\ge j$. A matrix is a {\em Fishburn} matrix if all of
its entries are non-negative integers such that neither row nor column
contains only zero entries. Let $\CMcal{F}_n$ be the set of Fishburn matrices
whose sum of entries equals $n$, then for any $M\in\CMcal{F}_n$, let 
\begin{align*}
  \mathsf{rowsum}_1(M)
  &:=\mbox{ the sum of entries in the first row of } M,\\
  \mathsf{ne}(M)
  &:=\mbox{ the number of weakly north-east cells of } M, \\
  \mathsf{mtr}(M)
  &:=\mbox{ the smallest index $i$ such that $(M_{1,i},\ldots,M_{i-1,i},M_{i,i})\ne (0,\ldots,0,1)$.}\\
  & \qquad \mbox{If no such index exists, then set } \mathsf{mtr}(M)=\dim(M).
\end{align*}
\begin{corollary}\label{C:fish}
There is a bijection between $\CMcal{F}_n$ and itself such that the
following symmetric distribution holds:
\begin{equation*}
(\mathsf{rowsum}_1,\mathsf{ne},\mathsf{mtr})M=
(\mathsf{rowsum}_1,\mathsf{mtr},\mathsf{ne})(\phi\circ \Phi\circ\phi^{-1})M,
\end{equation*}	
 where $\Phi$ is the bijection in Theorem \ref{T:5tuple} and
 $\phi:\A_n\rightarrow \CMcal{F}_n$ is the bijection from
 \cite[Theorem~3.6]{MR3904827}.
\end{corollary}
\begin{remark}		
The three Stirling statistics $\mathsf{rowsum}_1,\mathsf{ne}, \mathsf{mtr}$
are pairwise symmetric on $\CMcal{F}_n$. The fact that the pair
$(\mathsf{ne},\mathsf{mtr})$ is symmetric on $\CMcal{F}_n$ is a direct
consequence of Corollary~\ref{C:fish} and it is known from 
\cite{MR3904827,MR4011082,MR3388862} that the other two pairs
$(\mathsf{rowsum}_1,\mathsf{ne})$ and $(\mathsf{rowsum}_1,\mathsf{mtr})$
are also symmetric.
\end{remark}

\subsection{A conjecture about a symmetric quintuple equidistribution
on inversion sequences}
We pose a conjecture on a symmetric equidistribution of
Euler--Stirling statistics on inversion sequences, which is analogous to
Theorem \ref{T:5tuple} but with the two statistics $\ealm,\rpos$
(only defined on ascent sequences) being removed, and $\A_n$
(the set of ascent sequences) being replaced
by $\CMcal{I}_n$ (the set of inversion sequences).
\begin{conjecture}\label{OP:2}
There is a bijection $\Omega: \CMcal{I}_n\rightarrow \CMcal{I}_n$
such that for all $s\in \CMcal{I}_n$, 
\begin{equation*}
(\asc,\rep,\zero,\max,\rmin)s=(\asc,\rep,\zero,\rmin,\max)\Omega(s).
\end{equation*}
Consequently for all $\pi\in \mathfrak{S}_n$,
\begin{equation*}
  (\des,\iasc,\lmax,\lmin,\rmax)\pi
  =(\des,\iasc,\lmax,\rmax,\lmin)(b^{-1}\circ \Omega\circ b)(\pi),
\end{equation*}
where $b:\mathfrak{S}_n\rightarrow \CMcal{I}_n$ is a bijection due
to Baril and Vajnovszki (see Theorem 1 of \cite{MR3637993}).
\end{conjecture}

The validity of Conjecture~\ref{OP:2} has been verified
by Maple up to $n=10$. See \cite[Section~7]{JinSchlosser}
for a discussion on a possible approach to prove the conjecture.

%\begin{figure}
%\centering
%\includegraphics[width=0.5\textwidth]{plot}
%\caption{A plot of a function.}
%\label{fig:plot}
%\end{figure}



%\acknowledgements{}

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