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\documentclass[finalversion]{FPSAC2025}
\articlenumber{127}
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%\usepackage{enumerate}
\usepackage{amsmath,amssymb,amsfonts,
latexsym, amscd, epsfig, epic,color}
\usepackage{mathtools}
\usepackage{youngtab}
\usepackage{extarrows}
\usepackage{enumitem}
\usepackage{epstopdf}
%\usepackage{enumerate}
\usepackage{mathrsfs}
\usepackage{eucal}
%\usepackage{eufrak}
\usepackage{xspace}
%\usepackage{a4wide}
\usepackage{colordvi}
\usepackage{comment}
\usepackage{hyperref,eucal}
\usepackage{tabularx}
\usepackage{tikz}\usetikzlibrary{matrix,arrows}
\usepackage{ytableau} \usepackage{chemfig}
\usepackage{float}
\usepackage{youngtab}
\usepackage{mathrsfs}
\usepackage{caption}
\usepackage{epsfig}
\usepackage{stfloats}
\usepackage{booktabs}
\usepackage{changepage}
%% note that the class file already loads {amsmath, amsthm, amssymb}
%\theoremstyle{definition}
%\theoremstyle{example}
%\theoremstyle{remark}
%\theoremstyle{Theorem}
\newtheorem{thm}{Theorem}[section]
\newtheorem{lem}[thm]{Lemma}
\newtheorem{corollary}[thm]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[thm]{Definition}
\newtheorem{example}[thm]{Example}
\newtheorem{opq}[thm]{Open problem}
\newtheorem{remark}[thm]{Remark}
\def\W{\CMcal{W}}
\def\X{\CMcal{X}}
\def\O{\CMcal{O}}
\def\D{\CMcal{D}}
\def\E{\CMcal{E}}
\def\T{\CMcal{T}}
\def\M{\CMcal{M}}
\def\V{\CMcal{V}}
\def\R{\CMcal{R}}
\def\J{\CMcal{J}}
\def\A{\mathcal{A}}
\def\B{\mathcal{B}}
\def\S{\CMcal{S}}
\usepackage{lipsum}
\def\inv{\mathsf{inv}}
\def\quinv{\mathsf{quinv}}
\def\maj{\mathsf{maj}}
\def\ndes{\mathsf{ndes}}
\def\Des{\mathsf{Des}}
\def\des{\mathsf{des}}
\def\South{\mathsf{South}}
\def\ndes{\mathsf{ndes}}
\def\Ndes{\mathsf{Ndes}}
\def\csf{\mathsf{csf}}
\def\wdf{\mathsf{wdf}}
\def\dg{\mathsf{dg}}

\newcommand{\blambda}{\boldsymbol{\lambda}}
\newcommand{\bmu}{\boldsymbol{\mu}}
\newcommand{\bnu}{\boldsymbol{\nu}}
\newcommand{\GL}{\rm GL}
\newcommand{\bh}{\mathbf{h}}
\newcommand{\bg}{\mathbf{g}}
\newcommand{\UL}{\rm UL}
\newcommand{\U}{\rm U}
\newcommand{\cL}{\mathcal{L}}
\newcommand{\UT}{\rm UT}
\newcommand{\DT}{\rm DT}
\newcommand{\NT}{\rm NT}
\newcommand{\bbf}{\mathbb{F}}
\newcommand{\SCNPtt}{\mathbf{SC}_{\rm NPtt}}
\newcommand{\Set}{\mathsf{Set}}
\newcommand{\CQSym}{\mathsf{CQSym}}
\newcommand{\NSym}{\mathsf{NSym}}
\newcommand{\NCSym}{\mathsf{NCSym}}
\newcommand{\NCQSym}{\mathsf{NCQSym}}


\numberwithin{equation}{section}
\numberwithin{figure}{section}
% references alphabetically
% make relation clearer
\def\chk{\marginpar{\quad\qquad\textbf{check!}}}


%% define your title in the usual way
\title[Modified Macdonald polynomials and Mahonian statistics]{Modified Macdonald polynomials and Mahonian statistics}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
%% for more than 3 names, use the Oxford comma: One Author, Two Author, Red Author, \and Blue Author
\author[Emma Yu Jin and Xiaowei Lin]{\texorpdfstring{Emma Yu
Jin\thanks{\href{mailto:yjin@xmu.edu.cn}{yjin@xmu.edu.cn}}\addressmark{1}
\and Xiaowei Lin\thanks{\href{mailto:linxiaoweiqing@126.com}{linxiaoweiqing@126.com} }\addressmark{2}}{Emma Yu
Jin
and Xiaowei Lin}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1,2}School of Mathematical Sciences, Xiamen University, Xiamen 361005, China.}

%% put the date of submission here
\received{2024.11.09}

%% 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{We establish an equidistribution between the pairs of statistics 
$(\inv,\maj)$ and
$(\quinv,\maj)$ on any row-equivalency class $[\tau]$ where $\tau$ is a filling of a given Young diagram. In particular if $\tau$ is a filling of a rectangular diagram, the triples $(\inv,\quinv,\maj)$ and $(\quinv,\inv,\maj)$ have the same distribution over $[\tau]$.
Our main result affirms a conjecture proposed by Ayyer, Mandelshtam and Martin, thus presenting the equivalence between two refined formulas for the modified Macdonald polynomials.}

%% 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{modified Macdonald polynomials, bijections, inversions, queue inversions, the major index, equidistribution}

%% 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; i.e., [xx, yy] in the text

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\usepackage[backend=bibtex, giveninits]{biblatex}
\addbibresource{quinv_FPSAC.bib}
\appto\bibfont{\setlength{\emergencystretch}{1em}}
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\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 and main results}

%\lipsum[3]
{\em Macdonald polynomials} $P_{\lambda}(X;q,t)$ indexed by partitions are polynomials in infinitely many variables $X=\{x_1,x_2,\ldots\}$ with coefficients in the field $\mathbb{Q}(q,t)$ of rational functions of two variables $q$ and $t$. Several important classes of symmetric polynomials are well--studied specializations of Macdonald polynomials such as
Schur polynomials (when $q=t$), Hall--Littlewood polynomials (when $q=0$) and Jack polynomials (when $q=t^{\alpha}$ and let $t\rightarrow 1$).

Macdonald polynomials $P_{\lambda}(X;q,t)$ are defined as the unique basis for the ring of symmetric functions over the field $\mathbb{Q}(q,t)$ with orthogonal property and lower triangular property. The former is defined through the Hall scalar product and the latter by an expansion of $P_{\lambda}(X;q,t)$ with respect to monomial symmetric functions $m_{\lambda}(X)$. Since the coefficients in this expansion have nontrivial denominators, Macdonald introduced the {\em integral form} of $P_{\lambda}(X;q,t)$, denoted by $J_{\lambda}(X;q,t)$, which is defined as
\begin{align}\label{E:j1}
J_{\lambda}(X;q,t)=\sum_{\mu}K_{\mu\lambda}(q,t)s_{\mu}[X(1-t)],
\end{align}
where $f[X]$ denotes the plethystic substitution of $X$ into the symmetric function $f$, $s_{\mu}$ is the Schur function and $K_{\mu\lambda}(q,t)$ is the $q,t$-Kostka numbers.

Subsequently, another widely studied variant of Macdonald polynomials, called {\em modified Macdonald polynomials} $\tilde{H}_{\lambda}(X;q,t)$ was introduced by Garsia and Haiman \cite{GH93}. Let $\tilde{K}_{\lambda\mu}(q,t)=t^{n(\mu)}K_{\lambda\mu}(q,t^{-1})$ where $n(\lambda)=\sum_{i}(i-1)\lambda_i$. Then
\begin{align*}
\tilde{H}_{\lambda}(X;q,t)=\sum_{\mu}\tilde{K}_{\mu\lambda}(q,t)s_{\mu}(X).
\end{align*}
Haiman remarkably found that $\tilde{H}_{\lambda}(X;q,t)$ equals the Frobenius series of a space as the linear span of certain polynomials and their all partial derivatives \cite{Haiman01}. In parallel, the combinatorial investigation of modified Macdonald polynomials has been greatly promoted by the celebrated breakthrough on the surprising connections between $\tilde{H}_{\lambda}(X;q,t)$ and Mahonian statistics on fillings of Young diagrams due to {\parfillskip=0pt \par} \noindent Haglund, Haiman and Loehr \cite{HHL04}.

Before we state the combinatorial formula of $\tilde{H}_{\lambda}(X;q,t)$, let us review definitions of Mahonian statistics $\inv$, $\quinv$ and $\maj$ of fillings.
We represent a Young diagram in a French manner.
A partition $\lambda=(\lambda_1,\cdots,\lambda_k)$ of $n$ is sequence of positive integers such that $\lambda_i\ge \lambda_{i+1}$ for all $1\le i<k$ and $|\lambda|=\lambda_1+\cdots+\lambda_k=n$. Each $\lambda_i$ is called the $i$th part of $\lambda$ and $k$ is the length of $\lambda$, denoted by $\ell(\lambda)$. The Young diagram of $\lambda$, denoted by $\mathsf{dg}(\lambda)$, is an array of boxes with $\lambda_i$ boxes in the $i$th row from bottom to top, with the first box in each row left-justified. A box has coordinate $(i,j)$ if it is in the $i$th row from bottom to top and $j$th column from left to right. Let $\lambda'$ be the transpose of $\lambda$, that is, $\dg(\lambda')$ is obtained from $\dg(\lambda)$ by reflecting across the main diagonal (boxes with coordinate $(i,i)$).

A filling of $\mathsf{dg}(\lambda)$ is a function $\sigma :\mathsf{dg}(\lambda)\rightarrow \mathbb{P}$ ($\mathbb{P}$ is the set of positive integers), which assigns each box $u$ of $\mathsf{dg}(\lambda)$ to a positive integer $\sigma(u)$. We use $\South(u)$ to denote the box right below $u$. Let $\T(\lambda)$ denote the set of all fillings of $\mathsf{dg}(\lambda)$, and set
\begin{align*}
x^{\sigma}=\prod_{u\in \mathsf{dg}(\lambda)}x_{\sigma(u)}
\end{align*}
be the monomial of $\sigma$. A {\em descent} (or {\em non-descent}) of a filling $\sigma \in \T(\lambda)$ is a pair of entries $(\sigma(x),\sigma(\South(x))$ such that $\sigma(x)>\sigma(\South(x))$ (or $\sigma(x)\le \sigma(\South(x))$).
Define $\Des(\sigma)=\{x\in \mathsf{dg}(\lambda): (\sigma(x),\sigma(\South(x))$  is a descent$\}$ to be the {\em descent set} of $\sigma$ and $\des(\sigma)=|\Des(\sigma)|$.
Let $\mathsf{leg}(u)$ be the number of boxes strictly above $u$ in its column, then
\begin{align*}
\maj(\sigma)=\sum_{u\in\Des(\sigma)}({\mathsf{leg}}(u)+1)
\end{align*}
is called {\em the major index} of $\sigma$. %where $\chi(\A)=1$ if $\A$ is true; otherwise $\chi(\A)=0$.
%Let $\M(\lambda)$ be the set of monotonic fillings of $\lambda$. A column is a {\em descent column} (resp. {\em non-descent column}) if entries of this column are strictly increasing (resp. weakly decreasing) from bottom to top.


Given a filling $\sigma$, let $\hat{\sigma}$ be the filling obtained by adding a box with entry $0$ above the topmost box of each column of $\sigma$. A {\em queue inversion triple} of $\sigma$ is a triple $(a,b,c)$
of entries in $\hat{\sigma}$ such that  (as shown below on the left)
\begin{enumerate}
	\item $b$ and $c$ are in the same row and $c\in\sigma$ is to the right of $b$;
	\item $a$ and $b$ are in the same column such that $b$ is right below $a$;
	\item one of the conditions $a<b<c$, $b<c<a$, $c<a<b$ and $a=b\ne c$ is true.
\end{enumerate}
\begin{center}
	\begin{minipage}[H]{0.2\linewidth}
		\begin{ytableau}
		a  &\none[]
		\\
		b &\none[\dots] & c\\
	\end{ytableau}
	\end{minipage}
	\begin{minipage}[H]{0.1\linewidth}
		\begin{ytableau}
			a  &\none[\dots] & c
			\\
			b
			&  \none[]\\
		\end{ytableau}
	\end{minipage}
\end{center}
Let $\tilde{\sigma}$ be the filling obtained by adding a box with entry $\infty$ below the bottommost box of each column of $\sigma$. An {\em inversion triple} of $\sigma$ is a triple $(a,b,c)$ of entries in $\tilde{\sigma}$ satisfying the above (2)--(3) and (4), as shown above on the right.
\begin{enumerate}
	\setcounter{enumi}{3}
    \item $a$ and $c$ are in the same row and $c$ is to the right of $a$.
\end{enumerate}
Let quinv$(\sigma)$ and inv$(\sigma)$ be the numbers of queue inversion triples and inversion triples of $\sigma$, respectively.

We are now ready to present the combinatorial formula of $\tilde{H}_{\lambda}(X;q,t)$ by Haglund, Haiman and Loehr \cite{HHL04}:
\begin{align}\label{E:mmp1}
\tilde{H}_{\lambda}(X;q,t)=\sum_{\sigma\in \T(\lambda)}x^{\sigma}q^{\maj(\sigma)}t^{\inv(\sigma)}
\end{align}
Recently, Corteel, Haglund, Mandelshtam, Mason and Williams \cite{CHO22,CHO19} discovered a compact formula for $\tilde{H}_{\lambda}(X;q,t)$ which is summed over sorted tableaux and made a conjecture on an equivalent form of \Cref{E:mmp1}:
\begin{align}\label{E:mmp2}
\tilde{H}_{\lambda}(X;q,t)=\sum_{\sigma\in \T(\lambda)}x^{\sigma}q^{\maj(\sigma)}t^{\quinv(\sigma)}
\end{align}
This conjecture was confirmed by Ayyer, Mandelshtam and Martin \cite{AMM23} by proving that the RHS of \Cref{E:mmp2} satisfies certain orthogonal and triangular conditions which uniquely determine the modified Macdonald polynomials $\tilde{H}_{\lambda}(X;q,t)$.

Interestingly, a refinement of the equivalence between \Cref{E:mmp1,E:mmp2} was conjectured by Ayyer, Mandelshtam and Martin (\cite[Conjecture 10.3]{AMM23}).
Our main result is an affirmation of this conjecture; see \Cref{T:3},\Cref{E:T1}. As a bonus of our approach, we find the equidistribution \eqref{E:T2} between the triples $(\inv,\quinv,\maj)$ and $(\quinv,\inv,\maj)$ for all rectangular diagrams. To be precise, two fillings $\sigma,\tau$ of $\dg(\lambda)$ are called {\em row-equivalent}, denoted by  $\sigma\sim\tau$, if the multisets of entries in the $i$th row of $\sigma$ and $\tau$ are exactly the same for all $i$. The precise statement of our main result is the following.
\begin{thm}\label{T:3}
	 Let $[\sigma]$ denote the row-equivalent class of $\sigma$, then
\begin{align}\label{E:T1}
\sum_{\tau\in [\sigma]}q^{\maj(\tau)}t^{\inv(\tau)}=\sum_{\tau\in [\sigma]}q^{\maj(\tau)}t^{\quinv(\tau)}.
\end{align}	
If $\sigma$ is a filling of a rectangular diagram, then
\begin{align}\label{E:T2}
\sum_{\tau\in [\sigma]}q^{\maj(\tau)}t^{\inv(\tau)}u^{\quinv(\tau)}=\sum_{\tau\in [\sigma]}q^{\maj(\tau)}u^{\inv(\tau)}t^{\quinv(\tau)}.
\end{align}
\end{thm}
It is worth pointing out that the symmetric distribution \eqref{E:T2} is not true for arbitrary filling $\sigma$ and we provide such an example in \Cref{Tab:counterexample}.

Three subsets of $[\sigma]$, respectively with extreme values of the major index or (queue) inversion numbers are shown to satisfy \Cref{E:T1} by Bhattacharya, Ratheesh and Viswanath \cite{BRV23,R24}. Their proofs are bijective, which develop novel connections between different combinatorial models, maps and statistics such that
Gelfand--Tsetlin patterns, partitions overlaid patterns, box complementation \cite{BRV23} and charge and cocharge on words \cite{R24}.


We take a different approach, highlighting that the reverse operator and a column switch operator are sufficient to prove \Cref{T:3}. The rest of the extended abstract is organized as follows: In \Cref{S:2operators} the reverse operator and flip operator as the starting point of our proof are described. \Cref{S:mainproof} is devoted to proving \Cref{T:3}, with more details provided in \cite{JL24}.


%\lipsum[10]



%\section{Preliminaries and notations}\label{S:2}

%\lipsum[5]



%\lipsum[17]
\section{Reverse operator and flip operator}\label{S:2operators}
This section is concentrated on two operators, reverse operator and a queue inversion flip operator tailored to the statistic $\quinv$ given by Ayyer, Mandelshtam and Martin \cite{AMM23}. The latter was inspired by the column switch operator for the statistic $\inv$ by Loehr and Niese \cite{LN12}.

Both operators are related to a decomposition of the Young diagram of $\lambda$ into rectangles \cite{CHO22}. Each Young diagram $\dg(\lambda)$ is regarded as a concatenation of
maximal rectangles in a way that the heights of rectangles are strictly decreasing from left to right. For any $\sigma\in\T(\lambda)$, let $\sigma_i$ be the filling of the $i$th rectangle of $\dg(\lambda)$ and $\sigma=\sigma_1 \sqcup \cdots \sqcup \sigma_p$ where $p$ is the number of rectangles of $\dg(\lambda)$; see \Cref{F:b2} for an example.
\begin{figure}[ht]
	\centering
	\includegraphics[scale=0.4]{figure1.eps}
	\caption{\label{F:b2} A decomposition of $\sigma=\sigma_1 \sqcup\sigma_2\sqcup \cdots \sqcup \sigma_p$ (left), a filling $\sigma$ of a rectangle diagram (middle) and its reverse (right).}
\end{figure}

\begin{definition}[reverse operator]
For a partition $\lambda$ and $\sigma=\sigma_1 \sqcup \cdots \sqcup \sigma_p\in\T(\lambda)$, define $\sigma^r=\sigma_1^r\sqcup\cdots\sqcup \sigma_p^r$ as the reverse of $\sigma$ where the filling $\sigma^r$ is obtained by reversing the sequence of entries of each row; see \Cref{F:b2}.
\end{definition}
\noindent We adopt some notations from \cite{AMM23,O:24} to describe the flip operator.
Define $\CMcal{Q}(a,b,c)=1$ if $(a,b,c)$ is a queue inversion triple or an inversion triple; otherwise $\CMcal{Q}(a,b,c)=0$. %We denote by $\lambda_i$ the size of the $i$th row of $\dg(\lambda)$ and $\lambda'_i$ the size of the $i$th column of $\dg(\lambda)$. %and $\dg(\lambda')$ the transpose of $\dg(\lambda)$.
 We say that $i$ is {\em $\lambda$-compatible} if $\lambda'_i=\lambda'_{i+1}\ge 1$.
\begin{definition}[flip operator]\label{Def:2}
For $\sigma\in\T(\lambda)$ and $\lambda$-compatible $i$, let $t_i^{(r)}$ be the operator that acts on $\sigma$ by interchanging the entries $\sigma(r,i)$ and $\sigma(r,i+1)$.	For $1\le r,s\le \lambda'_i$, let
\begin{align*}
t_i^{[r,s]}:=t_i^{(r)}\circ t_i^{(r+1)}\cdots \circ t_i^{(s)}
\end{align*}
 denote the {\em flip operator} that swaps entries of boxes $(x,i)$ and $(x,i+1)$ for all $x$ with $r\le x\le s$. The {\em flip operator} $\rho_i^r$ is defined as follows: if columns $i$ and $i+1$ are identical in $\sigma$, then $\rho_i^r(\sigma)=\sigma$; otherwise, let $k$ be the maximal integer such that $\sigma(k,i)\ne \sigma(k,i+1)$ and $k\le r$. Let $h$ be maximal such that $h\le k$, $\sigma(h,i)\ne \sigma(h,i+1)$ and
 \begin{align*}
 \CMcal{Q}(\sigma(h,i),\sigma(h-1,i),\sigma(h-1,i+1))
 =\CMcal{Q}(\sigma(h,i+1),\sigma(h-1,i),\sigma(h-1,i+1)),
 \end{align*}
 where $\sigma(0,i)=\infty$ for all $i$. Define $\rho_i^r(\sigma)=t_i^{[h,k]}$, that is, reverse the pair of entries in every row between rows $h$ and $k$, columns $i$ and $i+1$. We call row $k$ (or $h$) {\em the starting row} (or {\em the ending row}) of $\rho_i^{r}$. For simplicity, we denote
 \begin{align*}
 \rho_i=\rho_i^{\lambda'_i}\quad\mbox{ and }\quad t_i=t_i^{(\lambda'_i)}.
 \end{align*}
 By definition $\rho_i^r\circ\rho_i^r(\sigma)=\sigma$, that is, $\rho_i^r$ is an involution on $\T(\lambda)$.
\end{definition}
\begin{remark}
We point out two differences between flip operator in \Cref{Def:2} and
the queue inversion flip operator introduced in \cite{AMM23,O:24}. First the latter always starts from the topmost row, that is $\rho_i$, while we allow the flip operator to begin from any row. Second, we add the condition $\sigma(h,i)\ne \sigma(h,i+1)$ to the terminating row $h$. These two modifications are intended to develop a precise relation between the change of $\quinv$, $\inv$ and $\CMcal{N}\des$ (see \Cref{thm3.4}), which contributes to the proof of \Cref{T:3}.
\end{remark}
\begin{example}
	Given a filling $\sigma$ as below, $\rho_1(\sigma)=t_1^{[3,5]}(\sigma)$ is generated as follows. Since $\young(93)$ is the topmost row with different entries, the flipping process
	$\rho_1$ starts from this row, i.e., row $5$ and continues until row $3$, where $\CMcal{Q}(3,5,8)=\CMcal{Q}(9,5,8)=1$. Consequently $\rho_1=t_1^{[3,5]}$.
	\begin{align*}
	\begin{ytableau}[] 3&3\\ 9 & 3\\ 2 & 5 \\ 3 & 9  \\ 5 & 8  \\ 9 &3
	\end{ytableau}
	\rightarrow
	\begin{ytableau}[] 3&3\\ 3 & 9\\ 2 & 5 \\ 3 & 9  \\ 5 & 8  \\ 9 &3
	\end{ytableau}
	\rightarrow
	\begin{ytableau}[] 3&3\\ 3 & 9\\ 5 & 2 \\ 3 & 9  \\ 5 & 8  \\ 9 &3
	\end{ytableau}
	\rightarrow
	\begin{ytableau}[] 3&3\\ 3 & 9\\ 5 & 2 \\ 9 & 3  \\ 5 & 8  \\ 9 &3
	\end{ytableau}
	\end{align*}
	
\end{example}

\section{Our proof strategy}\label{S:mainproof}
The proof of \Cref{T:3} is bijective, namely, for a partition $\lambda$, we will construct a bijection $\varphi:\T(\lambda)\rightarrow \T(\lambda)$ satisfying
\begin{align}\label{E:varphi1}
(\quinv,\maj)(\varphi(\sigma))=(\inv,\maj)(\sigma).
\end{align}
In particular, if $\dg(\lambda)$ is a rectangle, we prove that
\begin{align*}
(\inv,\quinv,\maj)(\varphi(\sigma))=(\quinv,\inv,\maj)(\sigma);
\end{align*}
otherwise, we find a filling $\sigma$ such that \Cref{E:T2} is no longer true (see \Cref{Tab:counterexample}).

The bijection $\varphi$ is a composition of two bijections associated with the flip operator $\rho_i^{k}$. The first one $\gamma$ is described in \Cref{pro1} (see below), which is reduced to the reverse operator if $\dg(\lambda)$ is a rectangle. Let
$\sigma=\sigma_1\sqcup\cdots \sqcup\sigma_p$, define
\begin{align}\label{eqthm3.6a}
\kappa(\sigma):= \sum_{i=1}^{p}(\quinv(\sigma_i)-\inv(\sigma_i^r)).
\end{align}
For any filling $\tau$, $\tau_1$ represents the leftmost rectangle in the decomposition of $\tau$. Define $\CMcal{N}\des(\tau)=(a_1,\ldots,a_k)$ where $a_i$ counts the number of non-descents in column $i$ of $\tau$ and set $\ndes(\tau)=a_1+\cdots+a_k$ be the number of non-descents of $\tau$.
\begin{thm}\label{pro1}
   There is a bijection $\gamma:\T(\lambda)\rightarrow \T(\lambda)$ satisfying $\gamma(\sigma)\sim \sigma$,
	\begin{align}\label{eqthm3.6b}
	\quinv(\gamma(\sigma))&=\inv(\sigma)+\kappa(\gamma(\sigma)),\\
	\label{eqthm3.6c}
	\maj(\gamma(\sigma))&=\maj(\sigma),\\
	\label{eqthm3.6d}\CMcal{N}\des(\sigma_1)&=\CMcal{N}\des((\gamma(\sigma)_1)^r)
	\end{align}
	and the topmost rows of $\sigma$ and $\gamma(\sigma)$ are reverse of each other. %Consequently, $\kappa(\sigma_1)=\kappa(\gamma(\sigma)_1)$.
\end{thm}
The second bijection $\theta:\T(\lambda)\rightarrow \T(\lambda)$ acts on each rectangle of the fillings independently and decreases the number of queue inversions by $\kappa(\gamma(\sigma))$ but preserves the major index, by which we find the desired bijection $\varphi$ with property \eqref{E:varphi1}.

\subsection{The proof of \texorpdfstring{{\Cref{T:3}}}{Theorem 1.1}} \label{S:6}
In this first step, we discuss the change of statistics $\quinv$, $\inv$ and $\maj$ by the reverse operator and the flip operator in \Cref{L:1,L:2}.

\begin{lem}\label{L:1}
	For $\lambda=(n^m)$ and $\sigma\in\T(\lambda)$, let $x_i$ be the number of non-descents in column $i$ of $\sigma$. Then, we have
	\begin{align}\label{eq5}
	\quinv(\sigma)-\inv(\sigma^r)=\inv(\sigma)-\quinv(\sigma^r)
	=\sum_{i=1}^{n} x_i(n-2i+1).
	\end{align}
\end{lem}

\begin{lem}\label{L:2}
	For a partition $\lambda$ and a $\lambda$-compatible $i$,
	let $\sigma\in\T(\lambda)$ and suppose that $\rho_i^{r}=t_i^{[\kappa_1,\kappa_2]}$. Let $\young(ab,cd)$ and $\young(st,uv)$ be parts of $\sigma$ such that $\young(cd)$ is the starting row $\kappa_2$ and $\young(st)$ is the ending row $\kappa_1$. Set $\sigma(0,i)=\infty$ and $\sigma(\lambda_i'+1,i)=0$ for all $i$.
	If $\CMcal{Q}(a,c,d)=\CMcal{Q}(b,c,d)=0$, then
	\begin{align*}
	\quinv(\sigma)+1=\quinv(\rho_i^{r}(\sigma)).
	\end{align*}
	Equivalently, if $\CMcal{Q}(a,c,d)=\CMcal{Q}(b,c,d)=1$, then
	\begin{align*}
	\quinv(\sigma)-1=\quinv(\rho_i^{r}(\sigma)).
	\end{align*}
	If $\CMcal{Q}(s,u,t)=\CMcal{Q}(s,v,t)=0$, then
	\begin{align*}
	\inv(\sigma)+1=\inv(\rho_i^{r}(\sigma)).
	\end{align*}
	Equivalently, if
	$\CMcal{Q}(s,u,t)=\CMcal{Q}(s,v,t)=1$, then
	\begin{align*}
	\inv(\sigma)-1=\inv(\rho_i^{r}(\sigma)).
	\end{align*}	
	For all cases, i.e., $\CMcal{Q}(a,c,d)=\CMcal{Q}(b,c,d)$ or $\CMcal{Q}(s,u,t)=\CMcal{Q}(s,v,t)$, we have
	\begin{align*}
	\maj(\sigma)=\maj(\rho_i^{r}(\sigma)).
	\end{align*}
\end{lem}
In the second step, with the help of \Cref{L:1,L:2}, we establish the involution $\phi_i$ in \Cref{thm3.4}, which relates the change of $\inv$ and $\quinv$ with the number of non-descent pairs.

\begin{thm}\label{thm3.4}
	For a partition $\lambda$ and a $\lambda$-compatible $i$, let $\sigma\in\T(\lambda)$ and $x_i$ be the number of non-descents in the $i$th column of $\sigma$.	
	Then there is an involution $\phi_i:\T(\lambda)\rightarrow \T(\lambda)$ such that $\phi_i(\sigma) \sim \sigma$, and for $\nu\in\{\inv,\quinv\}$,
	\begin{align}
	\label{E:majthm62}\maj(\phi_i(\sigma))&= \maj(\sigma), \\
	\label{E:quinvthm62}\nu(\phi_i(\sigma))&=\nu(\sigma)+x_{i+1}-x_i,\\
	\label{E:ndesthm62}\CMcal{N}\des(\phi_i(\sigma))&=(i,i+1)\circ \CMcal{N}\des(\sigma),
	\end{align}
	where $(i,i+1)\circ(\ldots x_i,x_{i+1}\ldots)=(\ldots x_{i+1},x_{i}\ldots)$.
\end{thm}

In the last step, building upon the properties  \Cref{E:majthm62}--\eqref{E:ndesthm62} of $\phi_i$, we construct the bijections $\theta$ and $\gamma$, thereby completing the proof of \Cref{T:3}.

\begin{thm}[second part of \Cref{T:3}]\label{thm8}
    If the diagram of $\lambda$ is a rectangle, then
 there is a bijection $\theta:\T(\lambda)\rightarrow \T(\lambda)$ such that $(\inv,\quinv,\maj)\sigma=(\quinv,\inv,\maj)\theta(\sigma)$.
\end{thm}
\begin{proof}
	For $\lambda=(n^m)$ and $\sigma\in\T(\lambda)$, let $x_i$ be the number of non-descents in column $i$. Then $\CMcal{N}\des(\sigma^r)=(x_n,\ldots,x_1)$. Define a bijection $\theta:\T(\lambda)\rightarrow \T(\lambda)$ as a product of $\phi_i$'s as follows.
	In the first step, apply the bijections $\phi_i$ for $i$ from $n-1$ to $1$ on $\sigma^r$, let $\tau_1=\phi_1\circ\cdots\circ\phi_{n-1}(\sigma^r)$. \Cref{thm3.4} ensures that the number of queue inversion triples is increased by
	\begin{align*}
	\nu(\tau_1)-\nu(\sigma^r)=\sum_{i=2}^{n}(x_1-x_i)=(n-1)x_1-\sum_{i=2}^{n}x_i
	\end{align*}
	for $\nu\in\{\inv,\quinv\}$ and $\CMcal{N}\des(\tau_1)=(x_1,x_n,\ldots,x_2)$. In the next step, we apply the bijections $\phi_i$ for $i$ from $n-1$ to $2$ on $\tau_1$ and let $\tau_2=\phi_2\circ\cdots\circ\phi_{n-1}(\tau_1)$, yielding that the number of queue inversion triples is increased by $(n-2)x_2-(x_3+\cdots+x_n)$ and $\CMcal{N}\des(\tau_2)=(x_1,x_2,x_n\ldots,x_3)$.
	Continue this process until the sequence $\CMcal{N}\des$ of the image becomes $(x_1,x_2,\ldots,x_n)$. Denote the resulting filling by $\theta(\sigma)$ and one sees that $\maj(\theta(\sigma))=\maj(\sigma)$ by \Cref{E:majthm62}. Further,
	\begin{align*}
	\nu(\theta(\sigma))&=\nu(\sigma^r)+\sum_{i=1}^{n-1}(n-i)x_i-\sum_{i=2}^{n}(i-1)x_i \\
	&=\nu(\sigma^r)+\sum_{i=1}^{n}x_i(n-2i+1).
	\end{align*}
	Compare with \Cref{eq5}, we conclude that
	$(\inv,\quinv)(\sigma)=(\quinv,\inv)(\theta(\sigma))$, as claimed.
\end{proof}
\begin{remark}\label{Tab:counterexample}
	In the table below, we present a row-equivalent filling class $[\sigma]$ of a non-rectangular diagram to disprove \Cref{E:T2} for arbitrary fillings.
	\begin{table}[H]
		\begin{adjustwidth}{-0cm}{0cm}
			\renewcommand\arraystretch{1.8}
			\begin{center}
				\resizebox{0.7\textwidth}{!}{
					\begin{tabular}{c| c c c c c c}%
						
						$[\sigma]$ & $\young(3,412,333)$ & $\young(3,421,333)$  &  $\young(3,124,333)$  &  $\young(3,142,333)$ & $\young(3,241,333)$& $\young(3,214,333)$\\ \hline
						
						$\maj$   & 2   &  2  &  2  & 2 &2 &2  \\ %\hline
						$\inv$   & 0   &  1  &  2  & 1 &2 &3  \\ %\hline
						
						
						$\quinv$  &3   & 2  & 2  & 1 &0&1  \\ %\hline
						
				\end{tabular}}
				%\caption{}
				\label{T:disp}
			\end{center}
		\end{adjustwidth}
	\end{table}
\end{remark}

We proceed by establishing \Cref{T:3} via \Cref{pro1} and \Cref{thm8}.
\begin{proof}[Proof of \Cref{T:3}]
For any $\sigma\in\T(\lambda)$, let $\tau=\gamma(\sigma)$, consider the rectangle decomposition of $\tau=\tau_1\sqcup\cdots \sqcup\tau_p$, let $\pi=\theta(\tau_1^r)\sqcup\cdots\sqcup \theta(\tau_p^r)$ and we shall see that $\varphi(\sigma)=\pi$ satisfying $\quinv(\pi)=\inv(\sigma)$ and $\maj(\pi)=\maj(\sigma)$. \Cref{thm8} assures that
\begin{align*}
\quinv(\pi)-\quinv(\tau)&=\sum_{i=1}^{p}(\quinv(\theta(\tau_i^r))-\quinv(\tau_i))=\sum_{i=1}^{p}(\inv(\tau_i^r)-\quinv(\tau_i))=-\kappa(\tau).
\end{align*}
In combination of $\quinv(\tau)-\inv(\sigma)=\kappa(\tau)$ by \Cref{pro1}, we conclude that $\quinv(\pi)$ equals $\inv(\sigma)$. Furthermore, $\maj(\pi)=\maj(\sigma)$ follows directly by \Cref{pro1} and \Cref{thm8}, as wished.
\end{proof}

\begin{example}\label{Exam:3}
	Let $\lambda=(7,7,5,5,5,2)$, a filling $\sigma$ of $\dg(\lambda)$ and the corresponding $\varphi(\sigma)$ are displayed as below left and right, respectively.
	\begin{align*}
	\begin{ytableau}[] 5&4\\ 9 & 3& 6& 1& 3\\ 2 & 5 & 9& 4& 8\\ 3 & 9 & 7&  3& 5  \\ 5 & 8 & 4&  6& 4 & 8&  7  \\ 9 &3 & 6  &5 & 2& 10  & 1
	\end{ytableau}	
	\hspace{7mm}
	\begin{ytableau}[]4 & 5\\  6& 3& 1& 3 &9\\ 8&4&5&9&2   \\ 3&3 &5 & 7 & 9\\ 7&8 &4 & 4 & 6& 8& 5\\ 10& 1 &6 & 2 & 5& 9& 3 \end{ytableau}
	\end{align*}
	One can easily check that $(\inv,\maj)(\sigma)=(\quinv,\maj)(\varphi(\sigma))=(40,33)$, but $\quinv(\sigma)\ne \inv(\varphi(\sigma))$ as $\quinv(\sigma)=32$ and $\inv(\varphi(\sigma))=34$.
\end{example}

\section{Discussion}
We propose some interesting open questions.
A {\em column strict filling} (CSF) of a diagram is a filling whose entries along each column are strictly decreasing from top to bottom. In other words, the major index of a column strict filling reaches its maximal value.


Bhattacharya, Ratheesh, and Viswanath \cite{BRV23} found the symmetric distribution \eqref{E:T2} for all the column strict fillings of an arbitrary diagram. That is, let $\mathsf{CSF}(\lambda)$ be the set of CSFs of the diagram of $\lambda$, then
\begin{align}\label{E:sym2}
\sum_{\tau\in [\sigma]\,\cap\,\mathsf{CSF}(\lambda)}t^{\inv(\tau)}u^{\quinv(\tau)}
=\sum_{\tau\in [\sigma]\,\cap\,\mathsf{CSF}(\lambda)}u^{\inv(\tau)}t^{\quinv(\tau)}.
\end{align}
\begin{opq}
The equation \eqref{E:sym2} is not directly seen from our bijection $\varphi$. It would be interesting to find out whether the bijection $\varphi$ is applicable to prove \Cref{E:sym2}.
\end{opq}

Combining \Cref{T:3} and results by Loehr and Niese \cite{LN12}, we are led to
\begin{align}\label{E:invq}
\sum_{\tau\in [\sigma]}t^{\maj(\tau')}=\sum_{\tau\in [\sigma]}t^{\inv(\tau)}=\sum_{\tau\in [\sigma]}t^{\quinv(\tau)}=\prod_{i=1}^{\ell(\lambda)}\genfrac{[}{]}{0pt}{}{a_{i,1}+\cdots+a_{i,N}}{a_{i,1},\ldots,a_{i,N}}_t,
\end{align}	
where $\tau'$ is obtained by transposing the filling $\tau$, the $i$th row of $\sigma$ consists of $a_{i,1}$ copies of $1$, $a_{i,2}$ copies of $2$, etc, and the rightmost one is a product of $t$-multinomial coefficients (see for instance \cite{Loehr,Stanley}).
\begin{opq}
Let $G(q,t,u)$ be the LHS of \Cref{E:T2}, can one derive a formula of $G(q,t,u)$ with the symmetric property $G(q,t,u)=G(q,u,t)$?
Is there a formula for \Cref{E:T1} as a natural generalization of \Cref{E:invq}?
\end{opq}


%The original motivation of \cite{LN12} is to bijectively prove an equality of a specialization of modified Macdonald polynomials in (\ref{E:mmp1}) via an inversion flip operator. It turns out that such operator is of central importance in deriving a compact formula for the modified Macdonald polynomials \cite{CHO22}.

%Let us recall a formula to explain the concept of Mahonian statistics \cite{LN12}. Let $\tau'$ be a filling obtained by transposing the filling $\tau$, then the polynomial of $\maj(\tau')$ over $\tau\in[\sigma]$ also equals (\ref{E:invq}), namely,
%\begin{align*}
%\sum_{\tau\in [\sigma]}t^{\inv(\tau)}=\sum_{\tau\in [\sigma]}t^{\maj(\tau')}.
%\end{align*}	
%Throughout the paper, any statistic whose distribution over $\T(\lambda)$ or $\T(\lambda')$ equals the distribution of $\inv$ on $\T(\lambda)$ is called a {\em Mahonian statistic}. Therefore, $\quinv,\inv,\maj$ are Mahonian statistics. Besides, we define $\chi(\A)=1$ if the statement $\A$ is true; and $\chi(\A)=0$ otherwise.


\acknowledgements{The first author was supported by the Austrian Research Fund FWF Elise-Richter Project V 898-N, and both authors are supported by the Fundamental Research Funds for the Central Universities, Project No. 20720220039. Both authors are supported by the National Nature Science Foundation of China (NSFC), Project No. 12201529.}

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