%% if you are submitting an initial manuscript then you should have submission as an option here
%% if you are submitting a revised manuscript then you should have revision as an option here
%% otherwise options taken by the article class will be accepted
\documentclass[finalversion]{FPSAC2018}
\articlenumber{12}
%% but DO NOT pass any options (or change anything else anywhere) which alters page size / layout / font size etc

%% note that the class file already loads {amsmath, amsthm, amssymb}

\usepackage{verbatim,amscd,color,graphics}
\usepackage{graphicx}
%\usepackage[colorlinks=true, pdfstartview=FitV, linkcolor=blue, citecolor=blue, urlcolor=blue]{hyperref}

\usepackage{tikz,ytableau}
%%%%%%%%%%%
\newcommand{\arxiv}[1]{\href{http://arxiv.org/abs/#1}{\texttt{arXiv:#1}}}

\numberwithin{equation}{section}
 
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{conjecture}[theorem]{Conjecture}

 
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}
 

% Dark red emphasis
\definecolor{darkred}{rgb}{0.7,0,0} % darkred color
\definecolor{darkgreen}{rgb}{0, .6, 0} % darkgreen color
\newcommand{\defn}[1]{{\color{darkred}\emph{#1}}} % emphasis of a definition

\usepackage[colorinlistoftodos]{todonotes}
\newcommand{\anne}[1]{\todo[size=\tiny,color=red!30]{#1 \\ \hfill --- Anne}}
\newcommand{\Anne}[1]{\todo[size=\tiny,inline,color=red!30]{#1
      \\ \hfill --- Anne}}
\newcommand{\Georgia}[1]{\todo[size=\tiny,inline,color=orange!30]{#1
      \\ \hfill --- Georgia}}
\newcommand{\Martha}[1]{\todo[size=\tiny,inline,color=blue!30]{#1
      \\ \hfill --- Martha}}
\newcommand{\Greta}[1]{\todo[size=\tiny,inline,color=green!30]{#1
      \\ \hfill --- Greta}}


\renewcommand{\L}{\mathsf{L}}
\newcommand{\R}{\mathsf{R}}
\newcommand{\De}{\mathsf{D}}
\newcommand{\ZZ}{\mathbb{Z}}  
\newcommand{\mm}{\mathsf{minimaj}} 
\newcommand{\maj}{\mathsf{maj}}
\newcommand{\wt}{\mathsf{wt}}
\renewcommand\read{\mathsf{read}}
\newcommand\calOP{\mathcal{OP}}
\newcommand{\mg}{\color{cyan}}

\newcommand\encircle[1]{%
  \tikz[baseline=(X.base)] 
    \node (X) [draw, shape=circle, inner sep=0] {\strut $#1$};}
           
%%%%%%%%%%%%%%%%%%%%%%%%%%%

%% define your title in the usual way
\title[A minimaj-preserving crystal structure on ordered multiset partitions]{A minimaj-preserving crystal structure on \\ordered multiset partitions}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author[Benkart, Colmenarejo, Harris, Orellana, Panova, Schilling, Yip]{
Georgia
Benkart\addressmark{1}\thanks{\href{mailto:benkart@math.wisc.edu}{benkart@math.wisc.edu}},
Laura Colmenarejo\addressmark{2}\thanks{\href{mailto:lcolme@yorku.ca}{lcolme@yorku.ca}},
Pamela E.
Harris\addressmark{3}\thanks{\href{mailto:peh2@williams.edu}{peh2@williams.edu}.
P. E. Harris was partially supported by NSF grant DMS--1620202.},
Rosa
Orellana\addressmark{4}\thanks{\href{mailto:Rosa.C.Orellana@dartmouth.edu}{Rosa.C.Orellana@dartmouth.edu}.
R. Orellana was partially supported by NSF grant DMS--1700058.},
Greta
Panova\addressmark{5}\thanks{\href{mailto:panova@math.upenn.edu}{panova@math.upenn.edu}.
G. Panova was partially supported by NSF grant DMS--1500834.},
Anne
Schilling\addressmark{6}\thanks{\href{mailto:anne@math.ucdavis.edu}{anne@math.ucdavis.edu}.
A. Schilling was partially supported by NSF grant  DMS--1500050.},\and
Martha
Yip\addressmark{7}\thanks{\href{mailto:martha.yip@uky.edu}{martha.yip@uky.edu}.
M. Yip was partially supported by Simons Collaboration grant 429920.}
}
%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics, University of Wisconsin-Madison\\ 
\addressmark{2}Department of Mathematics and Statistics,  York University\\
\addressmark{3}Mathematics and Statistics, Williams College\\
\addressmark{4}Mathematics Department, Dartmouth College\\
\addressmark{5}University of Pennsylvania and Institute for Advanced Study\\
\addressmark{6}Department of Mathematics, University of California at
Davis\\
\addressmark{7}Department of Mathematics, University of Kentucky}
%% put the date of submission here
\received{\today}

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

%% put your English abstract here, or comment this out if you don't have one yet
%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{
We provide a crystal structure on the set of ordered multiset partitions, which recently arose in
the pursuit of the Delta Conjecture.  This conjecture was stated by Haglund, Remmel and Wilson as a generalization 
of the Shuffle Conjecture. Various statistics on ordered multiset partitions arise in the combinatorial
analysis of the Delta Conjecture, one of them being the minimaj statistic, which is a variant of the
major index statistic on words. Our crystal has the property that the minimaj statistic is constant on connected 
components of the crystal. In particular, this yields another proof of the Schur positivity of the graded Frobenius 
series of the generalization $R_{n,k}$ due to Haglund, Rhoades and Shimozono of the coinvariant algebra $R_n$.
The crystal structure also yields a bijective proof of the equidistributivity of the minimaj statistic with the major index 
statistic on ordered multiset partitions.
}

%% put your French abstract here, or comment this out if you don't have one

%% put your keywords here, or comment this out if you don't have them yet
\keywords{Delta Conjecture, ordered multiset partitions, minimaj statistic, crystal bases, equidistribution of statistics}

%% 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{12-Team-Schilling.bib}
%% note the \printbibliography command at the end of the file which goes with these biblatex commands

\begin{document}

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

\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The Shuffle Conjecture~\cite{HHLRU.2005}, now a theorem due to Carlsson and Mellit~\cite{CM.2015},
provides an explicit combinatorial description of the bigraded Frobenius characteristic of the $S_n$-module of 
diagonal harmonic polynomials. Recently, Haglund, Remmel and Wilson~\cite{HRW.2015} introduced a generalization 
of the Shuffle Theorem, coined the Delta Conjecture. The Delta Conjecture involves two quasisymmetric functions
$\mathsf{Rise}_{n,k}(\mathbf{x};q,t)$ and $\mathsf{Val}_{n,k}(\mathbf{x};q,t)$, which have 
combinatorial expressions in terms of labelled Dyck paths. In this paper, we are only concerned with the specializations 
$q=0$ or $t=0$, in which case~\cite[Theorem 4.1]{HRW.2015} and~\cite[Theorem 1.3]{Rhoades.2016} show
\[
	\mathsf{Rise}_{n,k}(\mathbf{x};0,t) = \mathsf{Rise}_{n,k}(\mathbf{x};t,0) = 
	\mathsf{Val}_{n,k}(\mathbf{x};0,t) = \mathsf{Val}_{n,k}(\mathbf{x};t,0).
\]
It was proven in~\cite[Proposition 4.1]{HRW.2015} that
\begin{equation}
\label{equation.val}
	\mathsf{Val}_{n,k}(\mathbf{x};0,t) = \sum_{\pi\in \mathcal{OP}_{n,k+1}}  t^{\mm(\pi)} \mathbf{x}^{\wt(\pi)},
\end{equation}
where $\mathcal{OP}_{n,k+1}$ is the set of ordered multiset partitions of the multiset $\{1^{\nu_1},2^{\nu_2},\ldots\}$ 
into $k+1$ nonempty blocks and $\nu=(\nu_1,\nu_2,\ldots)$ ranges over all weak compositions of $n$.
The weak composition $\nu$ is also called the weight of $\pi$, denoted $\wt(\pi)=\nu$, and $\mathbf{x}^{\mathsf{wt}(\nu)} 
= x_1^{\nu_1}x_2^{\nu_2} \cdots$.
In addition, $\mm(\pi)$ is the minimum value of the major index of the set partition $\pi$ over all possible ways to 
order the elements in each block of $\pi$. The symmetric function $\mathsf{Val}_{n,k}(\mathbf{x};0,t)$ 
has an expansion as a sum of Schur functions with coefficients that are polynomials in $t$ with nonnegative integer 
coefficients~\cite{Wilson.2016,Rhoades.2016}.

In this paper, we provide a crystal structure on the set of ordered multiset partitions $\mathcal{OP}_{n,k}$.
Crystal bases are $q\to 0$ shadows of representations for quantum groups $U_q(\mathfrak g)$~\cite{Kashiwara.1990,
Kashiwara.1991}, though they can also be understood from a purely combinatorial 
perspective~\cite{Stembridge.2003,Bump.Schilling.2017}. In type $A$, the character of a connected crystal
component with highest weight element of highest weight $\lambda$ is the Schur function $\mathsf{s}_\lambda$.
Hence, having a crystal structure on a combinatorial set ($\mathcal{OP}_{n,k}$ in our case)
naturally yields the Schur expansion of the associated symmetric function. Furthermore, if the statistic
($\mm$ in our case) is constant on connected components, then the graded character can be computed using the crystal.

Haglund, Rhoades and Shimozono~\cite{HRS.2016} introduced a generalization $R_{n,k}$ for $k\leqslant n$ 
of the coinvariant algebra $R_n$, with $R_{n,n}=R_n$. Just as the combinatorics of $R_n$ is governed by permutations 
in $S_n$, the combinatorics of $R_{n,k}$ is controlled by ordered set partitions of $\{1,2\ldots,n\}$ with $k$ blocks.
The graded Frobenius series of $R_{n,k}$ is (up to a minor twist) equal to $\mathsf{Val}_{n,k}(\mathbf{x};0,t)$.
It is still an open problem to find a bigraded $S_n$-module whose Frobenius image is  
$\mathsf{Val}_{n,k}(\mathbf{x};q,t)$. Our crystal provides another representation-theoretic interpretation of
$\mathsf{Val}_{n,k}(\mathbf{x};0,t)$ as a crystal character.

Wilson~\cite{Wilson.2016} analyzed various statistics on ordered multiset partitions, including $\mathsf{inv}$,
$\mathsf{dinv}$, $\maj$, and $\mm$. In particular, he gave a Carlitz type bijection, which
proves equidistributivity of $\mathsf{inv}$, $\mathsf{dinv}$, $\maj$ on $\mathcal{OP}_{n,k}$.
Rhoades~\cite{Rhoades.2016} provided a non-bijective proof that these statistics are also equidistributed with
$\mm$. Using our new crystal, we can give a bijective proof of the equidistributivity of the 
$\mm$ statistic and the $\maj$ statistic on ordered multiset partitions.

This extended abstract is organized as follows. In Section~\ref{section.minimaj} we define ordered multiset partitions and 
the $\mm$ and $\maj$ statistics on them. In Section~\ref{section.bijection} we provide a bijection 
$\varphi$ from ordered multiset partitions to tuples of semistandard Young tableaux that will be used in 
Section~\ref{section.crystal} to define a minimaj-preserving crystal structure. We conclude in 
Section~\ref{section.equi} by showing that the $\mm$ and $\maj$ statistics are equidistributed using the same 
bijection $\varphi$.
We refer the reader to~\cite{BCHOPSY.2017} for further details and proofs of our results.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Ordered multiset partitions and two statistics}
\label{section.minimaj}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We consider \defn{ordered multiset partitions} of order $n$ with $k$ blocks.  
Given a weak composition $\nu = (\nu_1, \nu_2, \ldots)$ of $n$ into nonnegative integer parts,  
which we denote $\nu \models n$,  let $\mathcal{OP}_{\nu,k}$ be the set of partitions of the multiset 
$\{i^{\nu_i} \mid i \geqslant 1\}$  into $k$ nonempty ordered blocks, such that the elements within each block are distinct.  
The weak composition $\nu$ is also called the \defn{weight} $\wt(\pi)$ of 
$\pi \in \mathcal{OP}_{\nu,k}$. Let
\[
	\mathcal{OP}_{n,k}  =  \bigcup_{\nu \models n}  \mathcal{OP}_{\nu,k}.
\]

The \defn{minimaj order}, first defined in \cite{HRW.2015}, is a particular reading order for an ordered multiset partition $\pi = (\pi_1\mid \pi_2 \mid \ldots \mid \pi_k) \in \mathcal{OP}_{n,k}$ with blocks $\pi_i$, obtained as follows.
Start by writing the last block $\pi_k$ in increasing order, and continue to order the remaining blocks from right to left as follows.
Assume $\pi_{i+1}$ has been ordered and let $b_{i+1}$ be the leftmost letter of that block.  
If every element of $\pi_i$ is less than or equal to $b_{i+1}$, then write $\pi_i$ in the form $\pi_i = b_i \beta_i$ where 
$b_i \in \ZZ_{>0}$ and $\beta_i$ is an increasing sequence such that $b_i < \beta_i \leq b_{i+1}$. (For two sequences 
$\alpha,\beta$ of integers, we write $\alpha < \beta$ to mean that each element of $\alpha$ is less than every element of 
$\beta$.) Otherwise, $\pi_i$ can be written in the form $\pi_i = b_i \alpha_i \beta_i$ where $b_i\in \ZZ_{>0}$ and 
$\alpha_i, \beta_i$ are (possibly empty) sequences of increasing integers such that $\beta_i\leq b_{i+1} < b_i < \alpha_i$. 
Continue until every block has been ordered.  We consider the last block to be in the form $\pi_k = b_k \alpha_k$. 
See Example~\ref{example.pi}.
 
A sequence or word $w_1 w_2 \cdots w_n$ has a \defn{descent} in position $1\leqslant i<n$ if $w_i>w_{i+1}$.
Let $\pi \in \mathcal{OP}_{n,k}$ be in minimaj order. Observe that a descent 
is either between the largest and smallest elements of $\pi_i$
or between the last element of $\pi_i$ and the first element of $\pi_{i+1}$.   
Suppose that $\pi$ in minimaj order has descents in positions
\[
	\De(\pi) = \{d_1, d_1+d_2, \ldots, d_1+d_2 + \cdots + d_\ell\}
\]
for some $\ell \in [0,k-1]$.
The \defn{minimaj statistic} $\mm(\pi)$ of $\pi \in \mathcal{OP}_{n,k}$ as given by~\cite{HRW.2015} is
\begin{equation}
\label{equation.minimaj}
	\mm(\pi) = \sum_{d \in \De(\pi)} d  =  \sum_{j=1}^\ell (\ell+1-j) d_j.
\end{equation}

\begin{example}
\label{example.pi}
For $\pi = (157 \mid 24 \mid 56 \mid 468 \mid 13 \mid 123) \in \mathcal{OP}_{15,6}$, the minimaj order of $\pi$ is  
$\pi = (571 \mid 24 \mid 56 \mid 468 \mid 31 \mid 123)$.
With respect to the minimaj order, we have
\[
 	\begin{array}{lll}
	b_1 = 5, \alpha_1 =7, \beta_1 = 1 &\qquad b_2 = 2, \alpha_2 = \emptyset, \beta_2 =4 &\qquad
 	b_3 = 5, \alpha_3 = 6, \beta_3 =\emptyset \\ 
	b_4 = 4,  \alpha_4 = 68, \beta_4 =  \emptyset  &\qquad
 	b_5 = 3,  \alpha_5 = \emptyset, \beta_5 = 1 &\qquad b_6 = 1,  \alpha_6 = 23,  \beta_6 = \emptyset. 
	\end{array}
\]
The descents for the multiset partition  $\pi = (57.1 \mid 24 \mid 56. \mid 468. \mid 3.1 \mid 123)$  occur at positions 
$\De(\pi)=\{2,7,10,11\}$ and are designated with periods. Hence $\ell=4$, $d_1 = 2$, $d_2 = 5$, $d_3 = 3$, $d_4 = 1$ 
and $d_5 = 4$, and  $\mm(\pi) = 2 + 7 + 10 + 11 = 30$.  
\end{example}

To define the \defn{major index} of $\pi \in \mathcal{OP}_{n,k}$, we consider the word $w$ obtained by ordering each 
block $\pi_i$ in decreasing order, called the \defn{major index order}~\cite{Wilson.2016}.
The last element in the block $\pi_j$ is assigned the index $j$, and the remaining elements in $\pi_j$ are assigned the index $j-1$, for $j=1,\ldots, k$.  Writing the indices from left to right creates a word $v$. Then
\begin{equation}
\label{equation.maj}
	\maj(\pi) = \sum_{j \colon w_j>w_{j+1}} v_j.
\end{equation}

\begin{example}
Continuing Example~\ref{example.pi}, note that the major index order of $\pi = (157 \mid 24 \mid 56 \mid 468 \mid 13 
\mid 123) \in \mathcal{OP}_{15,6}$ is $\pi = (751 \mid 42 \mid 65 \mid 864 \mid 31 \mid 321)$. Writing the word $v$ 
underneath $w$ (omitting $v_0=0$), we obtain
\begin{equation*}
\begin{split}
	w &= 751 \mid 42 \mid 65 \mid 864\mid 31 \mid 321\\
	v &= 001 \mid 12 \mid 23 \mid 334 \mid 45 \mid 556,
\end{split}
\end{equation*}
so that $\maj(\pi) = 0+0+1+2+3+3+4+4+5+5=27$.
\end{example}

Note that throughout this section, we could have also restricted ourselves to ordered multiset partitions with 
letters in $\{1,2,\ldots, r\}$ instead of $\ZZ_{>0}$. That is, let $\nu=(\nu_1,\ldots,\nu_r)$ be a weak composition of $n$ and let
$\mathcal{OP}^{(r)}_{\nu,k}$ be the set of partitions of the multiset $\{i^{\nu_i} \mid 1\leqslant i \leqslant r\}$ into
$k$ nonempty ordered blocks, such that the elements within each block are distinct.  Let
\[
	\mathcal{OP}^{(r)}_{n,k}  =  \bigcup_{\nu \models n}  \mathcal{OP}^{(r)}_{\nu,k}.
\]
This restriction will be important when we discuss the crystal structure on ordered multiset partitions.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Bijection with tuples of semistandard Young tableaux}
\label{section.bijection}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section, we describe a bijection from ordered multiset partitions to tuples of semistandard Young tableaux
that allows us to impose a crystal structure on the set of ordered multiset partitions in Section~\ref{section.crystal}.

Recall that a \defn{semistandard Young tableau} $T$ is a filling of a (skew) Young diagram (also called the \defn{shape} of $T$)
with positive integers that weakly increase across rows and strictly increase down columns. The \defn{weight} of $T$ is 
the tuple $\wt(T)=(a_1,a_2,\ldots)$, where $a_i$ records the number of letters $i$ in $T$. The set of semistandard Young 
tableaux of shape $\lambda$, where $\lambda$ is a (skew) partition, is denoted by $\mathsf{SSYT}(\lambda)$. If we want to 
restrict the entries in the semistandard Young tableau from $\ZZ_{>0}$ to a finite alphabet $\{1,2,\ldots,r\}$, we denote the set by $\mathsf{SSYT}^{(r)}(\lambda)$.  

To state our bijection, we need the following notation. For fixed positive integers $n$ and $k$, assume 
$\mathrm{D}= \{d_1,d_1+d_2,\ldots,d_1+d_2+\cdots+d_\ell\} \subseteq \{1,2,\dots,n-1\}$ and 
$\mathrm{I} = \{i_1,i_1+i_2,\ldots,i_1+i_2+\cdots+i_\ell\} \subseteq\{1,2,\dots, k-1\}$ are sets of $\ell$ distinct elements each. 
Define $d_{\ell+1} := n-(d_1+\cdots +d_\ell)$ and $i_{\ell+1} := k - (i_1+\cdots +i_\ell)$.

\begin{proposition}
\label{P:biject}  
For fixed positive integers $n$ and $k$ and sets $\mathrm{D}$ and $\mathrm{I}$ as above, let
\[
	\mathrm{M(D,I)} = \{\pi \in \mathcal{OP}_{n,k} \mid  \De(\pi) = \mathrm{D}, \ 
	\text{and the descents occur in $\pi_i$ for $i \in \mathrm{I}$}\}.
\]
Then the following map is a weight-preserving bijection:
\begin{equation}
\begin{split}
	\varphi \colon \mathrm{M(D,I)} &\rightarrow \mathsf{SSYT}(1^{c_1}) \times \cdots 
	\times \mathsf{SSYT}(1^{c_\ell}) \times \mathsf{SSYT}(\gamma)\\
	\pi &\mapsto T_1 \times \cdots \times T_\ell \times T_{\ell+1}
\end{split}
\end{equation}
where
\begin{itemize}  
\item [{\rm (i)}] \ $\gamma = (1^{d_1-i_1}, i_1, \dots, i_{\ell+1})$ is of skew ribbon shape and $c_j = d_{\ell+2-j} - i_{\ell+2-j}$ for 
$1\leqslant j \leqslant \ell$.
\item [{\rm(ii)}] \ The skew ribbon tableau $T_{\ell+1}$ of shape $\gamma$ is constructed as follows:
\begin{itemize}  
\item [$\bullet$] The entries in the first column of $T_{\ell+1}$ beneath the first box are the first 
$d_1-i_1$ elements of $\pi$ in increasing order from top to bottom, excluding any $b_j$ in that range.
\item[$\bullet$] The remaining rows $d_1-i_1+j$ of $T_{\ell+1}$ for $1\leqslant j \leqslant \ell+1$ are filled with\newline
$b_{i_1+\dots + i_{j-1} + 1}, b_{i_1+\dots + i_{j-1} + 2}, \dots, b_{i_1+\dots + i_j}$.
\end{itemize} 
\item [{\rm(iii)}] The column tableau $T_j$ for $1\leqslant j \leqslant \ell$ of shape $1^{c_j}$ is filled with the elements of $\pi$ from the positions 
$d_1+d_2+\cdots + d_{\ell-j+1}+1$ through and including position $d_1+d_2+\cdots + d_{\ell-j+2}$, but excluding any 
$b_i$ in that range.
\end{itemize}
 \end{proposition}

Note that in item (ii), the rows of $\gamma$ are assumed to be numbered from bottom to top and are filled starting with row  $d_1-i_1+1$ and ending with row $d_1-i_1+\ell+1$ at the top.  
Also observe that since the entries of $\pi$ are mapped bijectively to the entries of $T_1 \times T_2 \times \cdots \times T_{\ell+1}$,  
the map $\varphi$ preserves the total weight $\wt(\pi)=(p_1,p_2,\ldots) \mapsto \, \wt(T)$, where $p_i$ is the number of entries $i$ in $\pi$ for $i \in \mathbb{Z}_{>0}$, so it can be restricted to a bijection
 \[
 	\varphi \colon \mathrm{M(D,I)}^{(r)} \rightarrow \mathsf{SSYT}^{(r)}(1^{c_1}) \times \cdots 
	\times \mathsf{SSYT}^{(r)}(1^{c_\ell}) \times \mathsf{SSYT}^{(r)}(\gamma),
\]
where $\mathrm{M(D,I)}^{(r)} = \mathrm{M(D,I)} \cap \mathcal{OP}^{(r)}_{n,k}$.

The next example illustrates the map $\varphi$. 
\begin{example}  
The ordered multiset partition $\pi = (124 \mid 45. \mid 3 \mid 46.1\mid 23.1\mid 1 \mid 25) \in \mathcal{OP}_{15,7}$ in 
minimaj order has the following data: $\ell=3$, $(d_1,d_2,d_3,d_4)=(5,3,3,4)$ and $(i_1,i_2,i_3,i_4)=(2,2,1,2)$.
Then 
\ytableausetup{boxsize=1.1em}
\[
	\pi = ({\color{blue}1}{\color{red}2}{\color{red}4} \mid {\color{blue}4}{\color{red}5}. \mid {\color{blue}3} \mid 
	{\color{blue}4}{\color{darkgreen}6}.{\color{orange}1}\mid {\color{blue}2}{\color{orange}3}.{\color{violet}1}\mid {\color{blue}1} 
	\mid {\color{blue}2}{\color{violet}5}) \mapsto
	\ytableaushort{{\color{violet}1},{\color{violet}5}} \times  \ytableaushort{{\color{orange}1},{\color{orange}3}} 
	\times \ytableaushort{{\color{darkgreen}6}} \times 
	\ytableaushort{\none\none {\color{blue}1}{\color{blue}2},\none\none {\color{blue}2},\none 
	{\color{blue}3}{\color{blue}4},{\color{blue}1}{\color{blue}4},{\color{red}2},{\color{red}4},{\color{red}5}}\;.
\]
\end{example}

\begin{proof}[Outline of the proof of Proposition \ref{P:biject}]
We explain how the map works for $\ell=0$. In this case, $\pi$ has no descents and the map $\varphi$ takes $\pi$ to the 
semistandard tableau $T=T_1$ of hook shape $\gamma=(1^{n-k},k)$ where the first row consists of the numbers 
$b_1,\ldots, b_k$, and the remainder of the first column is filled by the sequences $\beta_1,\beta_2,\ldots,\beta_{k-1},\alpha_k$
 in that order.

To reverse this map, suppose $T$ is a hook shape tableau with entries $b_1,\ldots, b_k$ in its first row and $b_1,t_1,\ldots, 
t_{n-k}$ in its first column.  The inverse $\varphi^{-1}$ maps $T$ to the set partition $\pi$ whose first block is 
$\pi_1=b_1\beta_1$ where $\beta_1 = t_1,\ldots,t_{m_1}$ such that $t_1 < \cdots <t_{m_1}\leq b_2$.  Continue in this way 
for each block $\pi_i$ and set the last block to $\pi_k = b_k\alpha_k$ where $\alpha_k = t_{m_{k-1}+1},\ldots, t_{n-k}$, so 
that $\pi$ is an ordered multiset partition with no descents.

In the general case for $\ell\geqslant 1$, we describe how $\varphi$ is reversed. The leftmost entry of each block of 
$\pi=\varphi^{-1}(T)$ is recovered from the entries of the skew ribbon tableau $T_{\ell+1}$ (excluding the bottom 
$d_1-i_1$ entries in the first column of $\gamma$), read from left to right.  Now, the remaining entries in the $(\ell+1)$-tuple 
of tableaux $T_1\times \cdots \times T_{\ell+1}$ are arranged in columns, and the last entry in each column is a member of 
$\alpha_j$ of some block $\pi_j = b_j\beta_j\alpha_j$.  The set $\mathrm{I}$ specifies that the bottommost entry in 
$T_{\ell+1-j}$ belongs to the block $\pi_{i_1+\cdots+i_j}$, and that entry is the position where a descent occurs in that block. 
Finally, since there are no more descent positions in $\pi$, there is a unique way to split up the remaining entries in 
$T_{\ell+1-j}$ by filling the blocks of $\pi$ from left to right, to complete the reconstruction of the ordered multiset partition 
$\pi = (\pi_1 \mid \pi_2 \mid \cdots \mid \pi_k)$.
\end{proof}
 
For a partition $\lambda$, the \defn{Schur function} $\mathsf{s}_\lambda(\mathbf{x})$ is defined as
\begin{equation}
\label{equation.schur}
	\mathsf{s}_\lambda(\mathbf{x}) = \sum_{T \in \mathsf{SSYT}(\lambda)} \mathbf{x}^{\wt(T)}.
\end{equation}
Similarly for $m \geqslant 1$, the \defn{$m$-th elementary symmetric function} $\mathsf{e}_m(\mathbf{x})$ is given by
\[
	\mathsf{e}_m(\mathbf{x}) = \sum_{1 \leqslant j_1 < j_2 < \cdots < j_m} x_{j_1} x_{j_2} \cdots x_{j_m}.
\]
As an immediate consequence of Proposition~\ref{P:biject}, we have the following symmetric function identity.
\begin{corollary}
\label{C:symexpand}   
Assume $\mathrm{D} \subseteq \{1,2,\dots,n-1\}$ and $\mathrm{I} \subseteq\{1,2,\dots, k-1\}$ are sets of $\ell$ 
distinct elements each and let $\mathrm{M(D,I)}$, $\gamma$ and $c_j$ for $1\leqslant j \leqslant \ell$ be as in 
Proposition~\ref{P:biject}. Then 
\[
	\sum_{\pi \in \mathrm{M(D,I)}} \mathbf{x}^{\wt(\pi)} 
	= \mathsf{s}_\gamma(\mathbf{x}) \ \prod_{j=1}^\ell \mathsf{e}_{c_j}(\mathbf{x}).
\]
\end{corollary}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Crystal on ordered multiset partitions}
\label{section.crystal}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Crystal structure}

Denote the set of words of length $n$ over the alphabet $\{1,2,\ldots,r\}$ by $\mathcal{W}^{(r)}_n$.
The set $\mathcal{W}_n^{(r)}$ can be endowed with an $\mathfrak{sl}_r$-crystal structure as follows.
The weight $\wt(w)$ of $w\in \mathcal{W}_n^{(r)}$ is the tuple $(a_1,\ldots,a_r)$, where $a_i$ is the number
of letters $i$ in $w$. The \defn{Kashiwara raising} and \defn{lowering operators}
\[
	e_i, f_i \colon \mathcal{W}_n^{(r)} \to \mathcal{W}_n^{(r)} \cup \{0\} \qquad \qquad \text{for $1\leqslant i <r$}
\]
are defined as follows. Associate to each letter $i$ in $w$ a closed
parenthesis ``$)$'' and to each letter $i+1$ in $w$
an open parenthesis ``$($''. Then $e_i$ changes the $i+1$ associated to the leftmost unmatched ``$($'' to an $i$; if there 
is no such letter, $e_i(w)=0$. Similarly, $f_i$ changes the $i$ associated to the rightmost unmatched ``$)$'' to an $i+1$;
if there is no such letter, $f_i(w)=0$.

For $\lambda$ a (skew) partition, the $\mathfrak{sl}_r$-crystal action on $\mathsf{SSYT}^{(r)}(\lambda)$ is induced
by the crystal on $\mathcal{W}_{|\lambda|}^{(r)}$, where $|\lambda|$ is the number of boxes in $\lambda$, by considering the 
row-reading word $\mathsf{row}(T)$ of $T\in \mathsf{SSYT}^{(r)}(\lambda)$, which is the word obtained from $T$ by 
reading the rows from bottom to top, left to right. 
In the same spirit, an $\mathfrak{sl}_r$-crystal structure can be imposed on 
\[
	\mathsf{SSYT}^{(r)}(1^{c_1},\ldots,1^{c_\ell},\gamma)
	:= \mathsf{SSYT}^{(r)}(1^{c_1}) \times \cdots \times \mathsf{SSYT}^{(r)}(1^{c_\ell}) \times \mathsf{SSYT}^{(r)}(\gamma)
\]
by concatenating the reading words of the tableaux in the tuple. This yields crystal operators
\[
	e_i,f_i \colon \mathsf{SSYT}^{(r)}(1^{c_1},\ldots,1^{c_\ell},\gamma) \to 
	\mathsf{SSYT}^{(r)}(1^{c_1},\ldots,1^{c_\ell},\gamma) \cup \{0\}.
\]
Via the bijection $\varphi$ of Proposition~\ref{P:biject}, this also imposes crystal operators on ordered
multiset partitions
\[
	\tilde{e}_i,\tilde{f}_i \colon \mathcal{OP}_{n,k}^{(r)} \to \mathcal{OP}_{n,k}^{(r)} \cup \{0\}
\]
as $\tilde{e}_i = \varphi^{-1} \circ e_i \circ \varphi$ and $\tilde{f}_i = \varphi^{-1} \circ f_i \circ \varphi$.

An example of a crystal structure on $\mathcal{OP}_{n,k}^{(r)}$ is given in Figure~\ref{figure.crystal}.

\begin{figure}
\centering
\scalebox{0.55}{
\begin{tikzpicture}[>=latex,line join=bevel,]
%%
\node (node_13) at (323.0bp,9.0bp) [draw,draw=none] {$\left(32 \mid 23\right)$};
  \node (node_14) at (114.0bp,305.0bp) [draw,draw=none] {$\left(1 \mid 123\right)$};
  \node (node_9) at (319.0bp,231.0bp) [draw,draw=none] {$\left(31\mid 12\right)$};
  \node (node_8) at (32.0bp,231.0bp) [draw,draw=none] {$\left(23 \mid 12\right)$};
  \node (node_7) at (360.0bp,157.0bp) [draw,draw=none] {$\left(31 \mid 13\right)$};
  \node (node_6) at (196.0bp,231.0bp) [draw,draw=none] {$\left(2\mid 123\right)$};
  \node (node_5) at (278.0bp,157.0bp) [draw,draw=none] {$\left(312\mid 2\right)$};
  \node (node_4) at (114.0bp,231.0bp) [draw,draw=none] {$\left(12\mid 23\right)$};
  \node (node_3) at (32.0bp,157.0bp) [draw,draw=none] {$\left(23 \mid 13\right)$};
  \node (node_2) at (196.0bp,157.0bp) [draw,draw=none] {$\left(3\mid 123\right)$};
  \node (node_1) at (114.0bp,157.0bp) [draw,draw=none] {$\left(123 \mid 3\right)$};
  \node (node_0) at (32.0bp,305.0bp) [draw,draw=none] {$\left(231\mid 1\right)$};
  \node (node_11) at (319.0bp,305.0bp) [draw,draw=none] {$\left(21 \mid 12\right)$};
  \node (node_10) at (323.0bp,83.0bp) [draw,draw=none] {$\left(31\mid 23\right)$};
  \node (node_12) at (196.0bp,305.0bp) [draw,draw=none] {$\left(21\mid 13\right)$};
  \draw [blue,->] (node_10) ..controls (323.0bp,62.872bp) and (323.0bp,42.801bp)  .. (node_13);
  \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  \pgfsetstrokecolor{strokecol}
  \draw (332.0bp,46.0bp) node {$1$};
  \draw [red,->] (node_6) ..controls (196.0bp,210.87bp) and (196.0bp,190.8bp)  .. (node_2);
  \draw (205.0bp,194.0bp) node {$2$};
  \draw [red,->] (node_9) ..controls (330.34bp,210.54bp) and (341.93bp,189.61bp)  .. (node_7);
  \draw (354.0bp,194.0bp) node {$2$};
  \draw [red,->] (node_8) ..controls (32.0bp,210.87bp) and (32.0bp,190.8bp)  .. (node_3);
  \draw (41.0bp,194.0bp) node {$2$};
  \draw [blue,->] (node_9) ..controls (307.66bp,210.54bp) and (296.07bp,189.61bp)  .. (node_5);
  \draw (287.0bp,194.0bp) node {$1$};
  \draw [red,->] (node_11) ..controls (319.0bp,284.87bp) and (319.0bp,264.8bp)  .. (node_9);
  \draw (328.0bp,268.0bp) node {$2$};
  \draw [red,->] (node_5) ..controls (290.44bp,136.54bp) and (303.17bp,115.61bp)  .. (node_10);
  \draw (287.0bp,120.0bp) node {$2$};
  \draw [blue,->] (node_0) ..controls (32.0bp,284.87bp) and (32.0bp,264.8bp)  .. (node_8);
  \draw (41.0bp,268.0bp) node {$1$};
  \draw [red,->] (node_4) ..controls (114.0bp,210.87bp) and (114.0bp,190.8bp)  .. (node_1);
  \draw (123.0bp,194.0bp) node {$2$};
  \draw [blue,->] (node_12) ..controls (196.0bp,284.87bp) and (196.0bp,264.8bp)  .. (node_6);
  \draw (205.0bp,268.0bp) node {$1$};
  \draw [blue,->] (node_7) ..controls (349.82bp,136.65bp) and (339.51bp,116.01bp)  .. (node_10);
  \draw (355.0bp,120.0bp) node {$1$};
  \draw [blue,->] (node_14) ..controls (114.0bp,284.87bp) and (114.0bp,264.8bp)  .. (node_4);
  \draw (123.0bp,268.0bp) node {$1$};
%
\end{tikzpicture}
}
\caption{The crystal structure on $\mathcal{OP}_{4,2}^{(3)}$. The $\mm$ of the connected components
are $2,0,1,1$ from left to right.
\label{figure.crystal}}
\end{figure}

\begin{theorem}
\label{theorem.crystal}
The operators $\tilde{e}_i, \tilde{f}_i$, and $\wt$ impose an $\mathfrak{sl}_r$-crystal structure on
$\mathcal{OP}_{n,k}^{(r)}$. In addition, $\tilde{e}_i$ and $\tilde{f}_i$ preserve the $\mm$ statistic.
\end{theorem}

\begin{proof}
By construction, the operators $\tilde{e}_i, \tilde{f}_i$, and $\wt$ impose an $\mathfrak{sl}_r$-crystal structure 
since $\varphi$ is a weight-preserving bijection. The Kashiwara operators $\tilde{e}_i$ and $\tilde{f}_i$ preserve
the $\mm$ statistic, since by Proposition~\ref{P:biject}, the bijection $\varphi$ restricts to $\mathrm{M(D,I)}^{(r)}$ 
which fixes the descents of the ordered multiset partitions in minimaj order.
\end{proof}

As shown in~\cite{BCHOPSY.2017}, the crystal operators $\tilde{e}_i, \tilde{f}_i$ can also be explicitly described
on $\mathcal{OP}_{n,k}$.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Schur expansion}

The \defn{character} of a crystal $B$ is defined as
$\mathrm{ch} B = \sum_{b\in B} \mathbf{x}^{\wt(b)}$.
Denote by $B(\lambda)$ the $\mathfrak{sl}_\infty$-crystal on $\mathsf{SSYT}(\lambda)$ defined above.
This is a connected highest weight crystal with highest weight $\lambda$, and the character $\mathrm{ch}B(\lambda) = \mathsf{s}_\lambda(\mathbf{x})$ is the Schur function defined in Equation~\eqref{equation.schur}.
Similarly, denoting by $B^{(r)}(\lambda)$ the $\mathfrak{sl}_r$-crystal on $\mathsf{SSYT}^{(r)}(\lambda)$,
its character is the Schur polynomial $\mathrm{ch}B^{(r)}(\lambda) = \mathsf{s}_\lambda(x_1,\ldots,x_r)$.
Let us define
\[
	\mathsf{Val}^{(r)}_{n,k}(\mathbf{x};0,t) = \sum_{\pi\in \mathcal{OP}^{(r)}_{n,k+1}}  t^{\mm(\pi)} \mathbf{x}^{\wt(\pi)},
\]
which satisfies $\mathsf{Val}_{n,k}(\mathbf{x};0,t) = \mathsf{Val}_{n,k}^{(r)}(\mathbf{x};0,t)$ for $r\geqslant n$,
where $\mathsf{Val}_{n,k}(\mathbf{x};0,t)$ is as in~\eqref{equation.val}.

As a consequence of Theorem~\ref{theorem.crystal}, we now obtain the Schur expansion of 
$\mathsf{Val}_{n,k}^{(r)}(\mathbf{x};0,t)$.

\begin{corollary}
We have
\[
	\mathsf{Val}_{n,k-1}^{(r)}(\mathbf{x};0,t) 
	= \sum_{\substack{\pi \in \mathcal{OP}^{(r)}_{n,k}\\ \tilde{e}_i(\pi) = 0 \;\; \forall \;1\leqslant i <r}} t^{\mm(\pi)} 
	\mathsf{s}_{\wt(\pi)}.
\]
\end{corollary}

\begin{example}
The crystal $\mathcal{OP}_{4,2}^{(3)}$, displayed in Figure~\ref{figure.crystal}, has four highest weight elements
with weights $(2,1,1)$, $(2,1,1)$, $(2,1,1)$, $(2,2)$ from left to right. Hence, we obtain the Schur expansion
\[
	\mathsf{Val}^{(3)}_{4,1}(\mathbf{x};0,t) = (1+t+t^2)\; \mathsf{s}_{(2,1,1)}(\mathbf{x}) + t \;\mathsf{s}_{(2,2)}(\mathbf{x}).
\]
\end{example}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Equidistributivity of the minimaj and maj statistics}
\label{section.equi}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section, we describe a bijection $\psi \colon \calOP_{n,k} \to \calOP_{n,k}$ in Theorem~\ref{theorem.bij OP}
with the property that $\mm(\pi) = \maj(\psi(\pi))$ for $\pi \in \calOP_{n,k}$. This proves the link between
$\mm$ and $\maj$ that was missing in~\cite{Wilson.2016}.
We can interpret $\psi$ as a crystal isomorphism, where $\calOP_{n,k}$ on the left is the $\mm$ crystal of
Section~\ref{section.crystal} and $\calOP_{n,k}$ on the right is viewed as a crystal of $k$ columns with elements
written in major index order.

The bijection $\psi$ (see Theorem~\ref{theorem.bij OP}) is the composition of $\varphi$ of Proposition~\ref{P:biject} with
a certain shift operator $\L$ (see Definition~\ref{definition.Lshift}).
When applying $\varphi$ to $\pi \in \calOP_{n,k}$, we obtain the tuple $T^\bullet=T_1 \times \cdots \times T_{\ell+1}$. 
We would like to view each column in the tuple of tableaux as a block of a new ordered multiset partition. However, note
that some columns could be empty, namely if $c_j=d_{\ell+2-j}-i_{\ell+2-j}$ in Proposition~\ref{P:biject} is zero for some 
$1\leqslant j \leqslant \ell$. For this reason, let us introduce the set of \defn{weak ordered multiset partitions} 
$\mathcal{WOP}_{n,k}$, where we relax the condition that all blocks need to be nonempty sets.

Define $\read(T^\bullet)$ as the weak ordered multiset partition whose blocks are obtained from $T^\bullet$ by reading 
the columns from the left to the right and from the bottom to the top.
Note that given $\pi= (\pi_1 | \pi_2| \cdots | \pi_k) \in \calOP_{n,k}$ in minimaj order, $\read(\varphi(\pi))$ is a weak
ordered multiset partition in major index order.  The map $\read$ is invertible.

\begin{example} 
\label{example.pi ex}
For $\pi = (1\mid 56.\mid 4.\mid 37.12\mid 2.1\mid 1\mid 34) \in \calOP_{13,7}$, in minimaj order, we have $\mm(\pi)=22$ 
and
\ytableausetup{boxsize=1.1em}
\[
	T^\bullet = \varphi(\pi) =  
	\ytableaushort{1,4} \times  \ytableaushort{1,2} 
	\times \ytableaushort{7} \times \emptyset \times 
	\ytableaushort{\none 13,\none 2,\none 3,\none 4,15,6}\,.
\]
Then $\pi'=\read(T^\bullet) = (4.1\mid 2.1\mid 7.\mid \emptyset \mid 6.1\mid 5.4.3.2.1 \mid 3)$.
\end{example}

\begin{definition}
\label{definition.Lshift}
We define the \defn{left shift operation} $\L$ on $\pi'\in \mathcal{I} = \{ \read(\varphi(\pi)) \mid \pi \in \mathcal{OP}_{n,k}\}$ 
as follows. Suppose $\pi'$ has $m \geqslant 0$ blocks $\pi_{p_m}',\ldots, \pi_{p_1}'$ that are either empty or have 
a descent at the end, and $1 \leqslant p_m < \cdots <  p_2 < p_1<k$. Set $\L(\pi') = \L^{(m)}(\pi')$,
where $\L^{(i)}$ for $0\leqslant i\leqslant m$ are defined as follows:
\begin{enumerate}
\item
Set $\L^{(0)}(\pi')=\pi'$.
\item
Suppose $\L^{(i-1)}(\pi')$ for $1\leqslant i \leqslant m$ is defined. By induction, the $p_i$-th block of $\L^{(i-1)}(\pi')$ is
$\pi'_{p_i}$. Let $S_i$ be the sequence of elements starting immediately to the right of block $\pi'_{p_i}$ in $\L^{(i-1)}(\pi')$ 
up to and including the $p_i$-th descent after the block $\pi_{p_i}'$. Let $\L^{(i)}(\pi')$ be the weak ordered multiset 
partition obtained by moving each element in $S_i$ one block to its left. 
\end{enumerate}
\end{definition}

\begin{example}
\label{example.pi ex2}
Continuing Example~\ref{example.pi ex}, we have 
$\pi'=(4.1\mid 2.1\mid 7.\mid \emptyset \mid {\color{blue}6}.{\color{blue}1}\mid {\color{blue}5}.{\color{blue}4}.{\color{blue}3}
.2.1 \mid 3)$, in major index order. We have $m=2$ with $p_2=3<4=p_1$, $S_1={\color{blue}61543}$, 
$S_2={\color{red}6154}$ and
\begin{equation*}
\begin{split}
	\L^{(1)}(\pi') &= (4.1\mid 2.1\mid 7. \mid {\color{red}6}.{\color{red}1}\mid {\color{red}5}.{\color{red}4}.3.\mid 2.1\mid 3),\\
	\L(\pi') = \L^{(2)}(\pi') &= (4.1\mid 2.1\mid 7.6.1 \mid 5.4.\mid 3.\mid 2.1\mid 3).
\end{split}
\end{equation*}
Note that $\maj(\pi')=28$, $\maj(\L^{(1)}(\pi'))=25$, and $\maj(\L(\pi')) = 22 = \mm(\pi)$.
\end{example}

\begin{proposition}
\label{proposition.L}
The left shift operation $\L \colon \mathcal{I} \to \mathcal{OP}_{n,k}$ is well defined.
\end{proposition}

\begin{definition}
\label{definition.Rshift}
We define the \defn{right shift operation} $\R$ on $\mu\in \mathcal{OP}_{n,k}$ in major index order as follows.  
Suppose $\mu$ has $m\geqslant 0$ blocks  $\mu_{q_1}, \ldots, \mu_{q_m}$ that have a descent at the end and 
$q_1 < q_2 < \cdots < q_m$. Set $\R(\mu) = \R^{(m)}(\mu)$,
where $\R^{(i)}$ for $0\leqslant i \leqslant m$ are defined as follows:
\begin{enumerate}
\item
Set $\R^{(0)}(\mu)=\mu$.
\item
Suppose $\R^{(i-1)}(\mu)$ for $1\leqslant i \leqslant m$ is defined. Let $U_i$ be the sequence of $q_i$ elements 
to the left of, and including, the last element in the $q_i$-th block of $\R^{(i-1)}(\mu)$.  Let $\R^{(i)}(\mu)$ be the 
weak ordered multiset partition obtained by moving each element in $U_i$ one block to its right.
Note that all blocks to the right of the $(q_i+1)$-th block are the same in $\mu$ and $\R^{(i)}(\mu)$.
\end{enumerate}
\end{definition}

\begin{example}
Continuing Example~\ref{example.pi ex2}, let $\mu = \L(\pi') = (4.1\mid 2.1\mid 7.6.1 \mid 5.4.\mid 3.\mid 2.1\mid 3)$.  
We have $m=2$ with $q_1=4<5=q_2$, $U_1=6154$, $U_2=61543$ and
\begin{equation*}
\begin{split}
	\R^{(1)}(\mu) &= (4.1\mid 2.1\mid 7. \mid 6.1 \mid 5.4.3.\mid 2.1\mid 3),\\
	\R(\mu) = \R^{(2)}(\mu) &= (4.1\mid 2.1\mid 7. \mid \emptyset \mid 6.1 \mid 5.4.3.2.1\mid 3),
\end{split}
\end{equation*}
which is the same as $\pi'$ in Example~\ref{example.pi ex2}.
\end{example}

\begin{proposition}
\label{proposition.R}
The right shift operation $\R$ is well defined and is the inverse of $\L$.
\end{proposition}

We extend the definition of the major index in a natural way to the set $\mathcal{WOP}_{n,k}$: the last element in a nonempty block $\pi_j'$ is assigned the index $j$, and the remaining elements in $\pi_j'$ are assigned the index $j-1$, for $j=1,\ldots, k$ where $\pi_j'\neq\emptyset$.

\begin{theorem} 
\label{theorem.bij OP}
Let $\psi \colon \calOP_{n,k}\rightarrow \calOP_{n,k}$ be the map defined by
\[
	\psi(\pi) = \L(\read(\varphi(\pi))) \qquad \text{for $\pi\in \calOP_{n,k}$ in minimaj order.}
\]
Then $\psi$ is a bijection that maps ordered multiset partitions in minimaj order to ordered multiset partitions in 
major index order. Furthermore, $\mm(\pi) = \maj(\psi(\pi))$.
\end{theorem}

\begin{proof}[Proof outline] The map $\psi$ is a bijection since each of the maps $\varphi$, $\read$, and $\L$ is invertible, so it remains to show that $\mm(\pi) = \maj(\psi(\pi))$ for $\pi \in \mathcal{OP}_{n,k}$ in minimaj order.

In the simple case that $\pi' = \read(\varphi(\pi))$ has no empty blocks and no descents at the end of any block, then $\L(\pi')=\pi'$, so that $\pi' = \psi(\pi)$. Then
\begin{equation}
\label{equation.base maj}
	\maj(\pi') = \sum_{j=1}^\ell (\ell+1-j) ( d_j - i_j -1) + \ell + \sum_{j=1}^\ell ( \ell+\eta_j-j),
\end{equation}
where $d_j,i_j,\eta_j = i_1 + \cdots + i_j$ are defined in Proposition~\ref{P:biject} for $\pi$.
Comparing with~\eqref{equation.minimaj},
\[
	\maj(\pi') = \mm(\pi) - \binom{\ell+1}{2} - \sum_{j=1}^\ell (\ell+1-j) i_j + \binom{\ell+1}{2} 
	+ \sum_{j=1}^\ell \eta_j = \mm(\pi),
\]
proving the claim.

In the general case that $\pi' = \read(\varphi(\pi))$ has a descent at the end of block $\pi'_p$ (respectively if $\pi'_p=\emptyset$), this will contribute an extra $p$ to the major index in~\eqref{equation.base maj} (respectively $p-1$). 
Hence, with the notation of Definition~\ref{definition.Lshift}, we have
$\maj(\pi') = \mm(\pi) + \sum_{i=1}^m p_i - e$,
where $e$ is the number of empty blocks in $\pi'$. 
Since $\psi(\pi) = \L(\pi')$, then noting that
we have for $1\leqslant i \leqslant m$
\[
	\maj(\L^{(i)}(\pi')) = \begin{cases}
	\maj(\L^{(i-1)}(\pi'))-p_i+1, & \text{if $\pi_{p_i}'=\emptyset$,}\\
	\maj(\L^{(i-1)}(\pi'))-p_i, &\text{if $\pi_{p_i}'$ has a descent at the end of its block},
	\end{cases}
\]
the claim follows. 
\end{proof}

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

\acknowledgements{Our work on this project began at the workshop \emph{Algebraic Combinatorixx 2} at  the Banff International 
Research Station (BIRS) 
in May 2017.  ``Team Schilling,''  as our group of authors is known, would like to extend thanks 
to the organizers of ACxx2,  to BIRS for hosting this workshop, and to the Mathematical Sciences Research Institute (MSRI) 
for sponsoring a follow-up meeting of some of the group members at MSRI in July 2017 supported by the National Science
Foundation under Grant No. DMS--1440140.
We would like to thank Meesue Yoo for early collaboration and Jim Haglund, Brendon Rhoades and Andrew Wilson for 
fruitful discussions. This work benefited from computations and experimentations in {\sc Sage}~\cite{sage}.
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% if you use biblatex then this generates the bibliography
%% if you use some other method then remove this and do it your own way
\printbibliography
%
%

\end{document}
