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\documentclass[finalversion]{FPSAC2023}
\articlenumber{84}
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%\usepackage{lmodern,bm,pxfonts}


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%  Environments 
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\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{conjecture}[theorem]{Conjecture}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{exercise}[theorem]{Exercise}

\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

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%%\usepackage{hyperref} %%\texorpdfstring{$(q,t)$}{(q,t)}-
%%\hypersetup{pdfauthor={Name}}


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%  MACROS for this particular document
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%Formatting
\newcommand{\newword}[1]{\emph{\textbf{#1}}}
\DeclareMathAlphabet{\mathpzc}{OT1}{pzc}{m}{it}

%MAPS
\newcommand{\Amap}{\mathpzc{A}}    %map frpm PF to Sn
\newcommand{\s}{\mathcal{S}}  % map from minimal coset reps to affine compositions


%SETS
\newcommand{\AS}{\widetilde{S}} %Affine Symmetric group
\newcommand{\dyckmn}{\operatorname{D}(m,n)} %% m-n dyck paths
\newcommand{\PF}{\operatorname{\mathsf{PF}}}   %%Parking functions
\newcommand{\PFmn}{\PF(m,n)} %%% for when PF_{mn} was used
\newcommand{\SSPF}{\operatorname{\mathsf{SSPF}}}   %%SSPF
%%%% need a macro with sub/super scripts if type differently
%%%% in various places
\newcommand{\SSPFr}{\SSPF^r(m,n)} %%?? {\SSPF_{(m,n)}^{\;r}}
\newcommand{\AC}{\operatorname{\mathcal{AC}}}  %% Affine compositions
\newcommand{\dyck}{\operatorname{{D(m,n)}}}

\newcommand{\Smin}{(S_{\bf{w}} \backslash \AS_m)^{\mathrm{min}}} %%% minimal length right coset representatives. 
\newcommand{\Snmin}{(S_{\bf{w}} \backslash \AS_m^n)^{\mathrm{min}}} %%% minimal length right coset representatives. 

%base fields
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\C}{\mathbb{C}}

%OPERATORS
\newcommand{\area}{\operatorname{\mathtt{area}}}
\newcommand{\coarea}{\operatorname{\mathtt{co-area}}}
\newcommand{\dinv}{\operatorname{\mathtt{dinv}}}
\newcommand{\codinv}{\operatorname{\mathtt{co-dinv}}}
\newcommand{\wt}{\operatorname{\mathtt{wt}}}
\newcommand\Des{\mathrm{Des}}
\DeclareMathOperator{\std}{\mathsf{std}}
\newcommand{\alg}{\operatorname{alg}}
\newcommand{\geom}{\operatorname{geom}}

% Special objects 
\newcommand{\D}{\mathsf{D}}  %% dyck path
\newcommand{\Hik}{\mathcal{H}} % hikita 
\DeclareMathOperator{\GL}{GL} 
\newcommand{\E}{\mathcal{E}} %elliptic hall
\newcommand{\EHA}{\mathcal{E}^{++}} %%e elliptic hall algebra
\newcommand{\DAHA}[1]{\mathbb{SH}(#1)^{++}} %% spherical DAHA
\newcommand{\pol}[1]{\mathrm{Pol}(#1)} %polynomial represenation


%%%%%%%%%%%%%%%%%%%%%%
% fancy comments
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\usepackage[normalem]{ulem}
\usepackage[colorinlistoftodos]{todonotes}
% 
\newcommand{\Jose}[1]{\todo[size=\tiny,inline,color=violet!30]{#1 \\ \hfill --- Jose}}
\newcommand{\Monica}[1]{\todo[size=\tiny,inline,color=blue!20]{#1 \\ \hfill --- Monica}}
\newcommand{\Nicolle}[1]{\todo[size=\tiny,inline,color=orange!30]{#1 \\ \hfill --- Nicolle}}
\newcommand{\TODO}[1]{\todo[size=\small,inline,color=red!60]{#1 \\ \hfill --- anyone}}

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%\input{../MainTexFiles-HigherRankCatalan/macros-Catalan.tex}


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%% math in pdf problems
%%\ensuremath{$(q,t)$} %%\texorpdfstring{$(q,t)$}{(q,t)}-

\title[
Semistandard Parking Functions and a Finite Shuffle Theorem
]{Semistandard Parking Functions and a Finite Shuffle Theorem}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
%\author[Optional shorter names]{Longer names me\thanks{\href{mailto:hello@world.c}{hello@world.c}. Longer names me was partially supported by Grant 2017.11.14.$\partial$\;supp.}\addressmark{1}, \and you\addressmark{2}}

%% then use \addressmark to match authors to institutions here
%\address{\addressmark{1}Department of This, University of That, There \\ \addressmark{2}School of Something, University of Something Else, Somewhere}

\author{Nicolle Gonz\'alez\thanks{\href{mailto:nicolles@berkeley.edu}{nicolles@berkeley.edu}.}\addressmark{1}, Jos\'e Simental\thanks{\href{mailto:simental@im.unam.mx}{simental@im.unam.mx}.}\addressmark{2}, \and Monica Vazirani\thanks{\href{mailto:vazirani@math.ucdavis.edu}{vazirani@math.ucdavis.edu}. M.Vazirani was partially supported
by Simons Foundation Grant 707426.}\addressmark{3}}

\address{\addressmark{1}Department of Mathematics,  University of California,  Berkeley, USA \\ \addressmark{2}Instituto de Matem\'aticas,  Universidad Nacional Aut\'onoma de M\'exico, CDMX, M\'exico\\ \addressmark{3}Department of Mathematics,  University of California,  Davis, USA}




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

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%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
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%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{ We introduce the higher rank rational $(q,t)$-Catalan
polynomials and prove these are equal to finite truncations of the
Hikita polynomial. We also generalize results of Gorsky-Mazin-Vazirani
and construct an explicit bijection between semistandard parking
functions and affine compositions. Using these results we prove a
finite analogue of the Rational Shuffle Theorem in the context of
spherical double affine Hecke algebras.}

%% 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{Catalan polynomial, Shuffle Theorem, elliptic Hall algebra, Hikita polynomial, DAHA, spherical DAHA.}

%% 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
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%% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex
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\usepackage[backend=bibtex]{biblatex}
\addbibresource{HigherRankCatalan_FPSAC.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
\emph{Catalan numbers} $C_n$ are some of the most ubiquitous quantities
throughout mathematics, naturally counting objects across a vast
array of fields. Their many generalizations, ranging from the so-called
\emph{rational Catalan numbers} $C_{(m,n)}$ to their bivariate $(q,t)$
counterpart $C_{(m,n)}(q,t)$, have played a central role in the deep
connections between $(q,t)$-combinatorics and important problems
arising in the K-theory of Hilbert schemes, the homology of torus
knots, the geometry of Gieseker varieties, and other areas of algebraic
geometry, topology, and representation theory.

A particularly important family of objects counted by Catalan numbers is the set of \emph{Dyck paths}, lattice paths in a square that do not cross the diagonal. Particular labelings of these paths give rise to \emph{parking functions}. %family of polynomials that arise as fillings over certain lattice paths.
 Originally introduced by Konheim and Weiss \cite{parking} in their study of hashing problems, parking functions were afterwards generalized by Armstrong, Loehr and Warrington  \cite{ALW} to the rational setting. %Correspondingly, these $(m,n)$-parking functions $\PFmn$ are counted by $C_{(m,n)}$. 
A new perspective on parking functions, inspired by Anderson \cite{anderson} and studied extensively by Gorsky-Mazin-Vazirani  \cite{GMV}, gives an explicit bijection between the set of parking functions $\PFmn$ and a certain set of affine permutations. More recently, motivated by the representation theory of quantizations of the Gieseker moduli space \cite{EKLS} Simental, jointly with Etingof-Krylov-Losev, generalized these constructions further by introducing the \emph{higher rank rational Catalan numbers} $C_{(m,n)}^{(r)}$ and \emph{higher rank semistandard parking functions} $\SSPFr$. As in the classical and rational cases, $|\SSPFr| = C_{(m,n)}^{(r)}$. %They proved that the semistandard parking functions index the affine spaces in an affine paving of a parabolic Springer fiber and that, as in the classical and rational cases, $|\SSPFr| = C_{(m,n)}^{(r)}$.

%Around the same time, Gorsky, Mazin and Vazirani provided a new perspective by constructing an explicit bijection between $\PFmn$ and the set of affine permutations \cite{GMV}. More recently, motivated by the representation theory of quantizations fo the Gieseker moduli space \cite{EKLS} Simental jointly with Etingof-Krylov-Losev generalized these constructions further by introducing the \emph{higher rank rational Catalan numbers} $C_{(m,n)}^{(r)}$ and \emph{higher rank semistandard parking functions} $\SSPFr$. As in the classical and rational cases, $|\SSPFr| = C_{(m,n)}^{(r)}$. %They proved that the semistandard parking functions index the affine spaces in an affine paving of a parabolic Springer fiber and that, as in the classical and rational cases, $|\SSPFr| = C_{(m,n)}^{(r)}$.
We generalize this further by extending the bijection given in \cite{GMV} to a bijection between $\SSPFr$
and affine compositions, which through an explicit standardization
procedure, allows us to define a ``$\dinv$'' statistic on these new sets
of objects. We use this statistic to introduce the \emph{higher rank rational
$(q,t)$-Catalan polynomial} $C_{(m,n)}^{(r)}(X;q,t)$. In \cite{GSV-2022} we further show how the $\dinv$ statistic  corresponds precisely to the dimension of
an associated affine space in an affine paving of a parabolic Springer
fiber.   

%We generalize this further and introduce the \emph{higher rank rational
%$(q,t)$-Catalan polynomial} $C_{(m,n)}^{(r)}(X;q,t)$. We then extend the
%bijection given in \cite{GMV} to a bijection between $\SSPF_{(m,n)}$
%and affine compositions, which through an explicit standardization
%procedure, allows us to define a ``$\dinv$'' statistic on these new sets
%of objects. In \cite{GSV-2022} we further show how this statistic on the
%semistandard parking functions corresponds precisely to the dimension of
%an associated affine space in an affine paving of a parabolic Springer
%fiber.   

Tying much of this theory together are the celebrated \emph{Shuffle Theorems}. In the classical setting, the Shuffle Theorem gives a combinatorial formula for the bigraded Frobenius character of the space of diagonal harmonics. While the Frobenius character of this representation was computed by Haiman geometrically in the early 2000's
\cite{Haiman01} and its combinatorial expression,
given as a sum over parking functions, was conjectured by
Haglund-Haiman-Loehr-Remmel-Ulyanov \cite{HHLRU} shortly thereafter, the conjecture was open for 15 years until Carlsson and Mellit \cite{CM}
%finally
proved it.
Unexpectedly arising in their studies of knot invariants via
Cherednik algebras, Gorsky and Negu\c{t} proposed a rational
generalization of the Shuffle Conjecture. In particular, they
conjectured that a certain \emph{elliptic Hall algebra} element acting
on $1$ inside the ring of symmetric functions gave rise to the
Frobenius character of a certain bigraded $S_n$ representation. The
combinatorial formula for this character, eponymously named the
\emph{Hikita polynomial}, had previously been computed by Hikita
\cite{Hikita}  as a
certain sum over $\PFmn$. Using similar methods as in the
classical case, Mellit  \cite{Mellit} successfully proved this
Rational Shuffle Theorem.

We connect our results to the Shuffle Theorems by first proving that
our higher rank rational $(q,t)$-Catalan polynomial corresponds
precisely to the truncation of the Hikita polynomial to a finite number
of variables. Then, using the fact that the elliptic Hall algebra
arises as the inverse limit of the spherical \emph{double affine Hecke
algebra} (DAHA) $\DAHA{r}$, we show there is an action of $\DAHA{r}$ on
the ring of symmetric polynomials whose action on $1$ results in these
higher rank rational $(q,t)$-Catalan polynomials. Since in the $r \to
\infty$ limit this agrees with the rational generalization of
Gorsky-Negu\c{t}, our construction gives a finite analogue of the Rational
Shuffle Theorem. 

\section{Background}

%\Nicolle{ Define: Dyck paths, parking functions, catalan numbers, rational catalan, hikita polynomial, rank of parking function, coarea, weight, regular dinv, anderson fillings}

Let $m,n \in \Z_{>0}$ satisfy $\gcd(m,n)=1$. An  $(m,n)$-\newword{Dyck path} is a lattice path consisting of only north and west steps from $(n,0)$ to $(0,m)$ bounded by the lines $x=0$, $y=0$, and $mx+ny=mn$. Denote by $\dyckmn$ the set of all $(m,n)$-Dyck paths and let $\mathcal{N}(\D)$ be the set of {\emph {north}} steps for any $\D \in \dyckmn$. 

Let $v_1, v_2, \dots, v_m$ and $u_1, u_2, \dots, u_n$ be the north and west steps of $\D \in \dyck$ read from $(n,0)$ to $(0,m)$.
We denote by $\dinv(\D)$
the number of \newword{diagonal inversions} of $\D$,
which consist of pairs $(u_r, v_s)$ over all $1\leq r \leq n$ and $ 1 \leq s \leq m$, such that both $u_r$ and $v_s$ are intersected by some line $\ell$ of slope $\frac{-m}{n}$ and $v_s$ lies southeast of $u_r$ in $\D$.
The \newword{area} of a Dyck path $\D \in \dyck$ is the number of lattice cells fully contained between the path $\D$ and the diagonal line $mx+ny=mn$. 
We set 
$\codinv(\D)= \frac{(m-1)(n-1)}{2} - \area(\D)$.

%\Monica{Ex1 has dinv=4, codinv=2 since $[03476]$ has 2 inversions}
\begin{example}
The Dyck path $\D$ in \cref{fig:5-4-PF} has $\area(\D) =2$ ,
$\dinv(\D)= 4$ and thus  $\coarea(\D)= \frac{(m-1)(n-1)}{2} -2 =4$,
$\codinv(\D)= \frac{(m-1)(n-1)}{2} - 4 = 2$.
\end{example}
%%$\dinv(\D, \varphi):= \frac{(m-1)(n-1)}{2}-\codinv(\D, \varphi)$.
%%% co- was not yet defined here. Jose: Do we need the co- at this point?

It is a well known fact that the number of $(m,n)$-Dyck paths
for coprime $m, n$ is given by the \emph{rational Catalan numbers}, $C_{(m,n)} := \frac{1}{n} {{n+m-1}\choose{m}}$. The \emph{$(q,t)$-Catalan number} is the bivariate deformation given by the polynomial
\begin{equation}
C_{(m,n)}(q,t):= \sum_{\D \in \dyckmn} q^{\area(\D)}t^{\dinv(\D)}.
\end{equation}

\subsection{Parking Functions and the Hikita Polynomial}

%%Introduced by Konheim and Weiss \cite{parking} and generalized by Armstrong, Loehr and Warrington in \cite{ALW},
%%\Monica{repetitive with Intro}

An \newword{$(m,n)$-parking function} is a pair $(\D, \varphi)$, where $\D$ is an $(m,n)$-Dyck path and $\varphi: \mathcal{N}(\D) \to \{1, \dots, m\}$ is a bijection that is strictly decreasing when reading upward on consecutive north steps. We will denote by $\PF(m,n)$ the set of $(m,n)$-parking functions.  

For example, in \cref{fig:std-unique}, the three rightmost diagrams correspond to all possible $(3,2)$-parking functions for the given Dyck path. 

Parking functions are in bijection with a certain set of affine permutations. To explain this bijection we recall the \newword{affine symmetric group},
\[
\AS_m:=\left\{ \sigma: \Z \to \Z \; \vert \; 
\sigma \text{ is a bijection, } \sigma(x+m)=\sigma(x)+m, \text{ and } \sum_{i=1}^m \sigma(i) = {{m+1}\choose{2}}\right\}.
\]
By the periodicity condition $\sigma \in \AS_m$ is completely determined by $\sigma(1), \dots, \sigma(m)$. We will use window notation and write $\sigma = [\sigma(1), \dots, \sigma(m)]$. We say $\sigma \in \AS_m$ is \newword{n-stable} if $\sigma(x+n) > \sigma(x)$ for all $x \in \Z$ and denote by $\AS_m^n$ the set of all n-stable affine permutations. 

%\Nicolle{add small example of an affine permutation where we introduce window notation}

In \cite{GMV}, the third author jointly with Gorsky and Mazin study a bijection (see \cref{sec:SSPF}) 
\begin{equation}\label{eq:PF-Amap}
\mathpzc{A}:  \PF(m,n) \to \AS_m^n,
\end{equation}
 %Using this, for $w = \mathpzc{A}(\D, \varphi)$, we set $\dinv(\D, \varphi)= \frac{(m-1)(n-1)}{2}-\codinv(\D, \varphi)$ and define $\codinv$ for $(\D, \varphi) \in \PF(m,n)$ as follows,
Using this, for $\sigma = \mathpzc{A}(\D, \varphi)$, we define
\begin{equation}\label{eq:codinv-PF}
\codinv(\D, \varphi):= | \{ (i,h) \in \{ 1, \dots, m\} \times \{ 1, \dots, n\} \; \vert \; \sigma(i+h) < \sigma(i) \} |.
\end{equation}
and set  $\dinv(\D, \varphi):= \frac{(m-1)(n-1)}{2}-\codinv(\D, \varphi)$.



\begin{figure}[ht]
\center
\begin{tikzpicture}
% pick an m and n 
 \pgfmathtruncatemacro{\m}{3}
  \pgfmathtruncatemacro{\n}{2}
%%%%%%% DONT TOUCH THIS PART
 \pgfmathtruncatemacro{\mi}{\m-1} %\mi=m-1 
    \pgfmathtruncatemacro{\ni}{\n-1} %\ni=n-1
    %
  \foreach \x in {0,...,\n} 
    \foreach \y in {0,...,\m}     
       {\pgfmathtruncatemacro{\label}{ - \m * \x - \n *  \y +\m*\n} 
       \node  at (\x-.2,\y-.2) [blue!60!white, scale=.7]{\label};} 
%generates the grid and anderson labels
\draw[dotted] (0,\m)--(\n,0); %draws the diagonal from (0,m) -- (n,0)
  \foreach \x in {0,...,\n} 
    \foreach \y  [count=\yi] in {0,...,\mi}    
      \draw (\x,\y)--(\x,\yi);
        \foreach \y in {0,...,\m}     
       \foreach \x  [count=\xi] in {0,...,\ni}          
         \draw (\x,\y)--(\xi,\y);
 %% Draw dyck path by hand below
 \draw[line width = 2pt,red] (0,3)--(0,1)--(1,1)--(1,0)--(2,0);
 % label the parking function, only modify the x coodinates ! 
\node at (0.5,2.5)[green!20!black]{2};
\node at (0.5,1.5) [green!20!black]{2};
\node at (1.5,0.5)[green!20!black] {2};
%%
\node at (1,-1) {$(\D,\varphi)$};
\end{tikzpicture}
%%
\qquad\qquad 
%%
\begin{tikzpicture}
% pick an m and n 
 \pgfmathtruncatemacro{\m}{3}
  \pgfmathtruncatemacro{\n}{2}
%%%%%%% DONT TOUCH THIS PART
 \pgfmathtruncatemacro{\mi}{\m-1} %\mi=m-1 
    \pgfmathtruncatemacro{\ni}{\n-1} %\ni=n-1
    %
  \foreach \x in {0,...,\n} 
    \foreach \y in {0,...,\m}     
       {\pgfmathtruncatemacro{\label}{ - \m * \x - \n *  \y +\m*\n} 
       \node  at (\x-.2,\y-.2) [blue!60!white, scale=.7]{\label};} 
%generates the grid and anderson labels
\draw[dotted] (0,\m)--(\n,0); %draws the diagonal from (0,m) -- (n,0)
  \foreach \x in {0,...,\n} 
    \foreach \y  [count=\yi] in {0,...,\mi}    
      \draw (\x,\y)--(\x,\yi);
        \foreach \y in {0,...,\m}     
       \foreach \x  [count=\xi] in {0,...,\ni}          
         \draw (\x,\y)--(\xi,\y);
 %% Draw dyck path by hand below
 \draw[line width = 2pt,red] (0,3)--(0,1)--(1,1)--(1,0)--(2,0);
 % label the parking function, only modify the x coodinates ! 
\node at (0.5,2.5)[green!20!black]{1};
\node at (0.5,1.5) [green!20!black]{3};
\node at (1.5,0.5)[green!20!black] {2};
%%
\node at (1,-1) {$\std(\D,\varphi)$};
\end{tikzpicture}
%%
\qquad\qquad 
%%
\begin{tikzpicture}
% pick an m and n 
 \pgfmathtruncatemacro{\m}{3}
  \pgfmathtruncatemacro{\n}{2}
%%%%%%% DONT TOUCH THIS PART
 \pgfmathtruncatemacro{\mi}{\m-1} %\mi=m-1 
    \pgfmathtruncatemacro{\ni}{\n-1} %\ni=n-1
    %
  \foreach \x in {0,...,\n} 
    \foreach \y in {0,...,\m}     
       {\pgfmathtruncatemacro{\label}{ - \m * \x - \n *  \y +\m*\n} 
       \node  at (\x-.2,\y-.2) [blue!60!white, scale=.7]{\label};} 
%generates the grid and anderson labels
\draw[dotted] (0,\m)--(\n,0); %draws the diagonal from (0,m) -- (n,0)
  \foreach \x in {0,...,\n} 
    \foreach \y  [count=\yi] in {0,...,\mi}    
      \draw (\x,\y)--(\x,\yi);
        \foreach \y in {0,...,\m}     
       \foreach \x  [count=\xi] in {0,...,\ni}          
         \draw (\x,\y)--(\xi,\y);
 %% Draw dyck path by hand below
 \draw[line width = 2pt,red] (0,3)--(0,1)--(1,1)--(1,0)--(2,0);
 % label the parking function, only modify the x coodinates ! 
\node at (0.5,2.5)[green!20!black]{1};
\node at (0.5,1.5) [green!20!black]{2};
\node at (1.5,0.5)[green!20!black] {3};
%%
\node at (1,-1.2) {};
\end{tikzpicture}
%%%
\qquad\qquad 
%%%%% comment for later - better to use scope and just one
%%%%% tikz for all 4 pictures
%%
\begin{tikzpicture}
% pick an m and n 
 \pgfmathtruncatemacro{\m}{3}
  \pgfmathtruncatemacro{\n}{2}
%%%%%%% DONT TOUCH THIS PART
 \pgfmathtruncatemacro{\mi}{\m-1} %\mi=m-1 
    \pgfmathtruncatemacro{\ni}{\n-1} %\ni=n-1
    %
  \foreach \x in {0,...,\n} 
    \foreach \y in {0,...,\m}     
       {\pgfmathtruncatemacro{\label}{ - \m * \x - \n *  \y +\m*\n} 
       \node  at (\x-.2,\y-.2) [blue!60!white, scale=.7]{\label};} 
%generates the grid and anderson labels
\draw[dotted] (0,\m)--(\n,0); %draws the diagonal from (0,m) -- (n,0)
  \foreach \x in {0,...,\n} 
    \foreach \y  [count=\yi] in {0,...,\mi}    
      \draw (\x,\y)--(\x,\yi);
        \foreach \y in {0,...,\m}     
       \foreach \x  [count=\xi] in {0,...,\ni}          
         \draw (\x,\y)--(\xi,\y);
 %% Draw dyck path by hand below
 \draw[line width = 2pt,red] (0,3)--(0,1)--(1,1)--(1,0)--(2,0);
 % label the parking function, only modify the x coodinates ! 
\node at (0.5,2.5)[green!20!black]{2};
\node at (0.5,1.5) [green!20!black]{3};
\node at (1.5,0.5)[green!20!black] {1};
%%
\node at (1,-1.2) {~};
\end{tikzpicture}
\caption{A $(3,2)$-semistandard parking function $(\D, \varphi)$ and
all $(3,2)$-parking functions for ${\color{red}{\D}}$, of which the second from the left 
is the unique
standardization $\std(\D, \varphi)$ determined by $\gamma$.
A box with upper right corner $(x,y)$ is marked by
${\color{blue}\gamma(x,y)}$.}
\label{fig:std-unique} \end{figure}


Gessel defined the \emph{fundamental quasisymmetric functions} for any subset $S \in \{1,\dots, m\}$ as the series
\begin{equation}
Q_{S} (X) := \sum_{\substack{i_1 \leq \cdots \leq i_m \\ j \in S \Rightarrow i_j < i_{j+1}}} x_{i_1}\dots x_{i_m}.
\end{equation}
These functions interpolate between elementary and complete symmetric functions and form a basis for the space of quasisymmetric functions 
in an infinite number of variables. 
\smallskip

Let $\area(\D,\varphi) := \area(\D)$ for any $(\D,\varphi) \in \PF(m,n)$. Then, the \newword{Hikita polynomial} is the $q,t$-symmetric function given by
\begin{equation}\label{eq:Hikita}
 \Hik_{(m,n)}(X ; q,t):= \sum_{(\D, \varphi) \in \PF(m,n) }q^{\area(\D, \varphi)}t^{\dinv(\D, \varphi)}Q_{\Des(\sigma^{-1})}(X), 
 \end{equation}
where $\Des(\sigma^{-1}):=\{ 1 \leq j \leq m \; \vert \; \sigma^{-1}(j+1) < \sigma^{-1}(j) \}$ denotes the \emph{descent set} of $\sigma^{-1}$ for $\sigma = \mathpzc{A}(\D, \varphi) \in \AS_m^n$.
When paired with the complete symmetric function $h_m(X)$ under the Hall inner product, the Hikita polynomial returns the rational $(q,t)$-Catalan number, that is $\langle \Hik_{(m,n)}, h_m \rangle = C_{(m,n)}(q,t).$


\subsection{Higher Rank Catalans and Semistandard Parking Functions}

 Motivated by the representation theory of quantizations of the Gieseker moduli space, in \cite{EKLS} Etingof, Krylov, Losev and the second author introduced the following generalization of parking functions.


\begin{definition} Let $r,m,n \in \Z_{>0}$.
A \newword{rank $r$ semistandard $(m,n)$-parking function} is a pair $(\D, \varphi)$ with $\D \in \dyckmn$ and a function $\varphi: \mathcal{N}(\D) \to \{ 1, \dots, r\}$ that is weakly decreasing when reading upward on consecutive north steps. Let $\SSPF^r(m,n)$ be the set of all rank $r$ semistandard $(m,n)$-parking functions.  
\end{definition}
In the special case when $r=m$, and $\varphi$ is a bijection, then the filling will automatically become standard, recovering the original set of $(m,n)$-\emph{parking functions} $\PF(m,n)$ considered in \cite{ALW}.

Given any $(\D, \varphi) \in \SSPF^r(m,n)$,  we let $\area(\D, \varphi):= \area(\D)$, define the \newword{weight} of $(\D, \varphi)$ as
the composition of $m$ given by
$\wt(\D,\varphi) := (|\varphi^{-1}(1)|,  |\varphi^{-1}(2)|, \dots, |\varphi^{-1}(r)|)$.
Let $\SSPF^r_\mathbf{w}(m,n)$ denote the set of all rank $r$ semistandard $(m,n)$-parking functions with weight $\mathbf{w}$. Note that for any $(\D, \varphi) \in \PF(m,n)$, we have $\wt(\D, \varphi) = (1^m)$.  

\begin{example} 
Consider $(\D,\varphi)$, the semistandard $(5,4)$-parking function in \cref{fig:5-4-PF} (left). Then  $(\D,\varphi)$ has rank $r=3$, $\area(\D,\varphi) = 2$, and $\wt(\D,\varphi) = (1,3,1)$. By comparison, the $(5,4)$-parking function in \cref{fig:5-4-PF} (right) has the same area but rank $5$ and weight $(1^5)$. 
\end{example}

In \cite[Theorem 2.28]{EKLS}, Etingof, Krylov, Losev and the second author defined the \newword{rank $r$ $(m,n)$-Catalan number} $C_{(m,n)}^{(r)}$ to be the cardinality of $\SSPF^r(m,n)$ and proved that when $m$ and $n$ are coprime,
\begin{equation} \label{eq:mn-Catalan-numbers}
C_{(m,n)}^{(r)} := |\SSPF^r(m,n)| = \frac{1}{n}\binom{nr+m-1}{m}.
\end{equation}


\section{Higher Rank Rational $(q,t)$-Catalan Polynomial}

As mentioned above, there exists a map $\Amap:  \PF(m,n) \to \AS_m^n$. We will recall this bijection and construct a generalization for
$\SSPF{(m,n)}$.
%%%$\SSPF_{m,n}$.
Denote by $[r]$ the set $\{ 1, \dots , r\} \subset \Z$. 

\begin{definition}
Let $m,r \in \Z_{>0}$. An \newword{$(m,r)$-affine composition} is a function $f: \Z \to \Z$ satisfying the following properties:
\begin{enumerate}
\item[(1)] $f(x+m) = f(x)+r$ for all $x \in \Z$.
\item[(2)] The set $f^{-1}[r]$ has exactly one element from each residue class mod $m$. 
\item[(3)] $\sum_{x \in f^{-1}[r]} x = {{m+1}\choose{2}}$.
\end{enumerate}
Denote by $\AC(m,r)$ the set of all $(m,r)$-affine compositions. 
See \cref{ex-window,ex-SSPF}.
\end{definition}
Just as for affine permutations, by the periodicity of condition (1) above, we use window notation and write $f= [ f(1), \dots , f(m)]_r$. The \newword{weight} of $f$ is given by 
\[\wt(f) = (|f^{-1}(1)|, |f^{-1}(2)|, \dots, |f^{-1}(r)|).\]
 Note that an affine composition of weight $(1^{m})$ is simply an affine permutation. As before, $f \in \AC(m,r)$ is \newword{n-stable} if $f(x+n) \geq f(x)$ for all $x \in \Z$. Then, let $\AC _{\mathbf{w}}(m,r)$ be the set of all $(m,r)$-affine compositions with weight $\mathbf{w}$ and denote by $\AC _{\mathbf{w}}^n(m,r)$ the subset of $\AC _{\mathbf{w}}(m,r)$ consisting of those affine compositions that are $n$-stable.

\begin{example}
\label{ex-window}
Let $m, r \in \Z_{> 0}$ and consider the $(m,r)$-affine composition $f \in \AC(m,r)$ that in window notation is $f = [1,1,\dots,1]_{r}$. Note that $f^{-1}[r] = [m]$ so that $f$ is indeed an affine composition of weight $(m, 0, \dots, 0) \in \Z_{\geq 0}^{r}$. We leave it to the reader to verify that $f$ is the function $f(x) = r\lfloor \frac{x-1}{m}\rfloor + 1$. 
\end{example}
%\Nicolle{add example of affine composition}

\subsection{Semistandard Parking Functions are Affine Compositions} \label{sec:SSPF}
We now describe 
$\Amap: \SSPF^r(m,n) \to \AC^n(m,r)$ which is a
 generalization of the map
$\Amap:  \PF(m,n) \to \AS_m^n$. %%  to %%$\SSPF(m,n)$.
 \smallskip

Given any $(\D, \varphi) \in \SSPF^{r}(m,n)$, index the north steps $v_1, \dots, v_m \in \mathcal{N}(\D)$ by their topmost coordinate, so that $v_j = (a_j, j)$ for each $1\leq j \leq m$ and in particular $a_m =0$. Define the function ${f}_\varphi: \Z \to \Z$ as follows: 
\begin{itemize}[leftmargin=3.5em]
\item[\bf{Step 1:}] Define $\gamma:\Z^2 \to \Z$ by $\gamma(x,y) = mn-mx-ny$. 

\item[\bf{Step 2:}] 
Set $\tilde{f}(\gamma(v_j)) = \varphi(v_j)$ for $j\in [m]$,
so that  $\tilde{f}^{-1}(i) = \gamma(\varphi^{-1}(i))$ for  $i \in [r]$.
%%%Set $\tilde{f}^{-1}(i) = \gamma(\varphi^{-1}(i))$ for each $1 \leq i \leq r$. This defines $\tilde{f}(\gamma(v_j))$ for $j\in [m]$.

\item[\bf{Step 3:}]  For any $x \not\in \{\gamma(v_j)\}_{1 \leq j \leq m}$, write $x = \gamma(v_j)+ pm$ for some $p \in \Z$ and $j \in [m]$ and set $\tilde{f}(x) := \tilde{f}(\gamma(v_j)) + pr$. 

\item[\bf{Step 4:}]  Let $k= \frac{n(m-1)-(m+1)}{2} - \sum_{j=1}^m a_j$, and set $f_\varphi(x) := \tilde{f}(x+k)$ for all $x \in \Z$. 
\end{itemize}

\noindent
Then we set $\Amap(\D,\varphi) = f_\varphi$.
Note that since $m,n$ are coprime then the set $\{ \gamma( v_j)\}_{1\leq j \leq m}$ contains exactly one element per residue class mod $m$. Thus $\tilde{f}$ is well-defined.   

\begin{theorem} \label{thm:SSPF-AC}
Let $r,m,n \in \Z_{>0}$ with $\gcd(m,n)=1$. The map
$\Amap: \SSPF^r(m,n) \to \AC^n(m,r)$
sending $(\D, \varphi)\mapsto f_\varphi$ is a weight-preserving bijection. Hence, given any $r$-part weak composition $\mathbf{w}$ of $m$, the construction above gives an isomorphism of sets, 
\[\Amap_\mathbf{w}: \SSPF_\mathbf{w}^r(m,n) \to \AC _{\mathbf{w}}^n(m,r).\]
\end{theorem}

In the special case when $r=m$ and $\mathbf{w} = (1^m)$, so that $\AC _{(1^m)}^n(m,m) = \AS^n_m$, \cref{thm:SSPF-AC}
recovers the map $\Amap:  \PF(m,n) \to \AS_m^n$ constructed in \cite{GMV}. For example, in \cref{fig:parking} we list  the affine permutations for all the $(3,2)$-parking functions. 

\begin{figure}[ht]
\center{
\begin{tikzpicture}
% pick an m and n 
 \pgfmathtruncatemacro{\m}{5}
  \pgfmathtruncatemacro{\n}{4}
%% DONT TOUCH THE REST
 \pgfmathtruncatemacro{\mi}{\m-1} %\mi=m-1 
    \pgfmathtruncatemacro{\ni}{\n-1} %\ni=n-1
    %
  \foreach \x in {0,...,\n} % n=5
    \foreach \y in {0,...,\m}     % m=7  
       {\pgfmathtruncatemacro{\label}{ - \m * \x - \n *  \y +\m*\n} % - m * \x - n *  \y +mn
      % \node   (\x\y) at (\x,\y){};
       \node  at (\x-.2,\y-.2) [blue!60!white, scale=.7]{\label};} 
%generates the grid and anderson labels
\draw[dotted] (0,\m)--(\n,0); %draws the diagonal from (0,m) -- (n,0)
  \foreach \x in {0,...,\n} % n=5
    \foreach \y  [count=\yi] in {0,...,\mi}   % m-1=6  
    %\draw (\x\y)--(\x\yi);
      \draw (\x,\y)--(\x,\yi);
        \foreach \y in {0,...,\m}     % m=7  
       \foreach \x  [count=\xi] in {0,...,\ni}   % n-1=4
        %\draw (\x\y)--(\xi\y);
         \draw (\x,\y)--(\xi,\y);
 % draw Dyck path by hand
\draw[line width = 2pt,red] (0,5) --(0,3) -- (1,3)--(1,1)--(2,1)--(2,0)--(4,0); 
% label the parking function, only modify the x coodinates ! 
\node at (.5,4.5) [green!20!black]{1};
\node at (.5,3.5) [green!20!black]{2};
\node at (1.5,2.5) [green!20!black]{2};
\node at (1.5,1.5) [green!20!black]{2};
\node at (2.5,0.5)[green!20!black]{3};
%% i know Nicolle said not to touch but adding color  \Monica{}
\node at (-.2,4.8)[red, scale=.7]{0};
\node at (-.2,3.8)[red, scale=.7]{4};
\node at (0.8,2.8)[red, scale=.7]{3};
\node at (0.8,1.8)[red, scale=.7]{7};
\node at (1.8,0.8)[red, scale=.7]{6};
\end{tikzpicture}
\qquad \qquad
\begin{tikzpicture}
% pick an m and n 
 \pgfmathtruncatemacro{\m}{5}
  \pgfmathtruncatemacro{\n}{4}
%% DONT TOUCH THE REST
 \pgfmathtruncatemacro{\mi}{\m-1} %\mi=m-1 
    \pgfmathtruncatemacro{\ni}{\n-1} %\ni=n-1
    %
  \foreach \x in {0,...,\n} % n=5
    \foreach \y in {0,...,\m}     % m=7  
       {\pgfmathtruncatemacro{\label}{ - \m * \x - \n *  \y +\m*\n} % - m * \x - n *  \y +mn
      % \node   (\x\y) at (\x,\y){};
       \node  at (\x-.2,\y-.2) [blue!60!white, scale=.7]{\label};} 
%generates the grid and anderson labels
\draw[dotted] (0,\m)--(\n,0); %draws the diagonal from (0,m) -- (n,0)
  \foreach \x in {0,...,\n} % n=5
    \foreach \y  [count=\yi] in {0,...,\mi}   % m-1=6  
    %\draw (\x\y)--(\x\yi);
      \draw (\x,\y)--(\x,\yi);
        \foreach \y in {0,...,\m}     % m=7  
       \foreach \x  [count=\xi] in {0,...,\ni}   % n-1=4
        %\draw (\x\y)--(\xi\y);
         \draw (\x,\y)--(\xi,\y);
 % draw Dyck path by hand
\draw[line width = 2pt,red] (0,5) --(0,3) -- (1,3)--(1,1)--(2,1)--(2,0)--(4,0); 
% label the parking function, only modify the x coodinates ! 
\node at (.5,4.5) [green!20!black]{1};
\node at (.5,3.5) [green!20!black]{3};
\node at (1.5,2.5) [green!20!black]{2};
\node at (1.5,1.5) [green!20!black]{4};
\node at (2.5,0.5)[green!20!black]{5};
\end{tikzpicture}}
\caption{A semistandard $(5,4)$-parking function with rank $r=3$ (left), and its standardization (right).
A box with upper right corner $(x,y)$ is marked by
${\color{blue}\gamma(x,y)}$.} \label{fig:5-4-PF}
\end{figure}

\begin{example}
\label{ex-SSPF}
Set $m =5, n=4$ and $r = 3$. Let $(\D, \varphi)$ be the rank $3$ semistandard $(5,4)$-parking function in \cref{fig:5-4-PF} (left). Note that $\wt(D, \varphi) = (1,3,1)$. From the figure we can read that
\[
\gamma(\varphi^{-1}(1)) = \{0\}, \qquad \gamma(\varphi^{-1}(2)) = \{3,4,7\}, \qquad \gamma(\varphi^{-1}(3)) = \{6\}.
\]
Thus, setting $\tilde{f}(\gamma(v_j)+5p) := \tilde{f}(\gamma(v_j)) + 3p$ we find that $\tilde{f}$ is the following map:
%\Monica{re-do colors to match figure}
\[
\begin{array}{cccccc|ccccc|cccccc}
  \color{blue}\cdots &  \color{blue}-4 & \color{blue} -3 &  \color{blue} -2 & \color{blue}-1 & \color{red}0 & \color{blue}1 &  \color{blue} 2& \color{red}3 & \color{red}4 & \color{blue}5 & \color{red}6 &\color{red}7 &  \color{blue}8 & \color{blue}9 & \color{blue} 10 &  \color{blue}\cdots  \\
\cdots &  -3 & -4 &  -1 &-1 & \mathbf{1} & 0 &-1 &\mathbf{2} & 
\mathbf{2} & 4 &  \mathbf{3} &  \mathbf{2} & 5 & 5 &7 & \cdots 
\end{array}
\]
\begin{comment}
\[
\begin{array}{cccccc|ccccc|cccccc}
 \cdots & -4 & -3 &  -2 &-1 & \color{red}0 &1 &  2& \color{red}3 & \color{red}4 &5 & \color{red}6 &\color{red}7 & 8 &9 & 10 & \cdots  \\
 \color{blue}\cdots &   \color{blue}-3 & \color{blue} -4 &  \color{blue} -1 & \color{blue}-1 &  \color{blue}1 &  \color{blue}0 & \color{blue}-1 & \color{blue}2 &  \color{blue}2 & \color{blue}4 &  \color{blue}3 &  \color{blue}2 &  \color{blue}5 &  \color{blue}5 & \color{blue}7 &  \color{blue}\cdots 
\end{array}
\]
\end{comment}
Now since the x-coordinates of the north steps $\{v_j=(a_j,j)\}_{1\leq j\leq 7}$ are given by $a_1=2$, $a_2=a_3=1$, and $a_4=a_5=0$, it follows that $k=1$. Thus, setting $f_\varphi(x) = \tilde{f}(x+1)$ we find that $f_\varphi = [-1,2,2,4,3]_{3}$, thus $\wt(f_\varphi) = \wt(\tilde{f}) = (1,3,1)$, as expected. Notice that by construction $f_\varphi^{-1}[3] = \{\gamma(v_j)-1\}= \{-1,2,3,5,6\}$ which, modulo $5$, is equivalent to the set
%%%$\{0,\dots, 4\}$.
$\{1,\dots, 5\}$.
Summing the entries in this set we get ${{6}\choose{2}}$, hence $f_\varphi$ is indeed a $(5,3)$-affine composition, which by inspection can be seen to be $4$-stable. 
\end{example}


%\begin{example} Set $m=7,n=5$, and $ r=10$.
%Let $(\D, \varphi)$ be the rank $10$ semistandard $(7,5)$-parking function in \cref{fig:7-5-PF} with $\wt(\D, \varphi)= (1,2,1,0,0,0,1,1,0,1)$. From the entries of the parking function we can see that 
%\begin{align*}
%\gamma (\varphi^{-1}(1) ) = \{3\}, && \gamma (\varphi^{-1}(2) )&= \{6,11\}, && \gamma (\varphi^{-1}(3) ) = \{16\}, \\
%\gamma (\varphi^{-1}(7) ) = \{8\}, && \gamma (\varphi^{-1}(8) ) &= \{0\}, && \gamma (\varphi^{-1}(10) ) = \{5\}.
 %\end{align*}
%Thus, setting $\tilde{f}(b+7a) := \tilde{f}(b) + 10a$ we find that $\tilde{f}$ is the following map:
%\[
%\begin{array}{cccc|ccccccc|ccccccc|ccccccc|ccccccc|ccccccc|cc}
 %\cdots & -2 & -1 &  \color{red}0 &1 & 2 & \color{red}3 &4 &\color{red}5 & \color{red}6 &7 & \color{red}8 &9 & 10 & \color{red}11 &12 & 13 &14 & 15 & \color{red}16 & \cdots  \\
 %\color{blue}\cdots &   \color{blue}0 & 	 \color{blue}-8 &  \color{blue}8 & \color{blue}-3 &  \color{blue}-17 &  \color{blue}1 & \color{blue}-8 & \color{blue}10 &  \color{blue}2 & \color{blue}18 &  \color{blue}7 &  \color{blue}-7 &  \color{blue}11 &  \color{blue}2 & \color{blue}20 &  \color{blue}12 & \color{blue}28 &  \color{blue}27 & \color{blue} 3 &  \color{blue}\cdots 
%\end{array}
%\]
%Now since the x-coordinates of the north steps $\{v_j=(a_j,j)\}_{1\leq j\leq 7}$ are given by $a_1=a_2=a_3=2$, $a_4=a_5=1$, and $a_6=a_7=0$, it follows that $k=3$. Thus, setting $f_\varphi(x) = \tilde{f}(x+3)$ we find that $f_\varphi = [-8,10,2,18,7,-7,11]_{10}$, thus $\wt(f_\varphi)= \wt(\tilde{f}) = (1,2,1,0,0,0,1,1,0,1)$, as expected.
%Notice that by construction $f_\varphi^{-1}[10] = \{\gamma(v_j)-3\}= \{-3,0,2,3,5,8,13\}$ which, modulo $7$, is equivalent to the set $\{0,\dots, 6\}$. Summing the entries in this set we get ${{8}\choose{2}}$, hence $f_\varphi$ is indeed a $(7,10)$-affine composition, which by inspection can be seen to be $5$-stable. 
%\end{example}



%\begin{example}
%Set $m=7,n=5$, and $ r=3$.
%Let $(\D, \varphi)$ be the rank $3$ semistandard $(7,5)$-parking function in \cref{fig:7-5-PF} with $\wt(\D, \varphi)= (3,3,1)$. Reading directly from the figure it is clear that $ \tilde{f}(\{0,6,11\}) = 1,\tilde{f}(\{3,8,16\}) =2,$ and $\tilde{f}(5) =3$. Thus, $\tilde{f}=[-1,-4,2,-2,3,1,4]_3$ and since $k=3$ it follows that $f_\varphi = [-2,3,1,4,2,-1,5]_3$.
%\end{example}



\subsection{Standardization and Diagonal Inversions}

The bijection in \cref{thm:SSPF-AC} allows us to compute $\codinv$ via Equation \eqref{eq:codinv-PF}, provided that $(\D,\varphi) \in \PF(m,n)$. Thus, in order to define the diagonal inversions for any semistandard parking function we utilize a standardization map from $\SSPF(m,n)~\to~\PF(m,n)$. 

For ${\bf{w}}=(w_1,\dots, w_r)$, an $r$-part weak composition of $m$, let $S_{\bf{w}} = S_{w_1} \times \dots \times S_{w_r}$ be the parabolic subgroup of $\AS_m$ indexed by $\bf{w}$. Denote by
%%%$S_{\bf{w}} \backslash \AS_m$ %%%make macro
$\Smin$
the set of minimal length right coset \emph{representatives}
and let $\Snmin = \Smin \cap \AS_m^n$ . 

\begin{proposition} Let $m,r \in \Z_{>0}$.
Given ${\bf{w}}=(w_1,\dots, w_r)$, an $r$-part weak composition of $m$, let $_{\bf{w}}f$ be the affine composition of weight $\bf{w}$ given by $_{\bf{w}}f:=[\underbrace{1,\dots, 1}_{w_1},
\underbrace{2\dots, 2}_{w_2},\dots,
\underbrace{r,\dots,r}_{w_r}]_r$.
Then the map 
\[\s_{\bf{w}}: 
\Smin
\to \AC_{\bf{w}}(m,r)\]
sending $\sigma \mapsto \;_{\bf{w}}f \circ \sigma$ is a bijection. Moreover, $\s_{\bf{w}}$ preserves $n$-stability and thus restricts to a bijection $\s_{\bf{w}}^n: \Snmin
\to \AC_{\bf{w}}^n(m,r)$ for any $n \in \Z_{>0}$ coprime to $m$.
\end{proposition}


\begin{definition}
For any $(\D,\varphi) \in \SSPF_{\bf{w}}^r(m,n)$ the \newword{standardization} of $(\D,\varphi)$ is the parking function $\std(\D,\varphi) \in \PF(m,n)$ given by,
\[ \std(\D,\varphi) := \Amap^{-1} \s_{\bf{w}}^{-1} \Amap_{\bf{w}} (\D,\varphi).\]
\end{definition}


\begin{theorem}
For any $(\D,\varphi) \in \SSPF_{\bf{w}}^r(m,n)$, its standardization $\std(\D,\varphi)$ is the $(m,n)$-parking function  $(\D, \alpha)$ with $\alpha:~ \mathcal{N}(\D) \to \{1, \dots, m\}$ the unique bijection satisfying:
\begin{itemize}
\item[(1)] $\alpha(\varphi^{-1}(a))<\alpha(\varphi^{-1}(b))$ for all $ 1\leq a<b \leq r$, and
\item[(2)] $\alpha(v_i)< \alpha(v_j)$ whenever $\varphi(v_i)= \varphi(v_j)$ and $\gamma(v_i) < \gamma(v_j)$.
\end{itemize}
\end{theorem}
As seen in \cref{fig:std-unique} conditions $(1)$ and $(2)$
ensure a unique standard representative of a semistandard parking
function  $(\D,\varphi)$.

Hence, for any rank $r$ semistandard $(m,n)$-parking function $(\D, \varphi)$ we set
\begin{equation}
\dinv(\D, \varphi) : = \dinv (\std(\D,\varphi)).
\end{equation}

\begin{example}
Consider the rank $r=3$ semistandard $(5,4)$-parking function $(\D,\varphi)$ in \cref{fig:5-4-PF} (left) with preimages $\varphi^{-1}(1) = \{v_5\}$, $\varphi^{-1}(2) = \{v_2,v_3,v_4\}$, and $\varphi^{-1}(3) = \{v_1\}$.  Since $\gamma(v_2) = \color{red} 7$, $\gamma(v_3)= \color{red} 3$, and $\gamma(v_4) =\color{red} 4$, then $\alpha(v_5) < \alpha(v_3)<\alpha(v_4)<\alpha(v_2)< \alpha(v_1)$. Hence $\alpha$ is the unique map from $\mathcal{N}(\D) \to \{1, \dots, 5\}$ given in \cref{fig:5-4-PF} (right).
\end{example}

\begin{example}
Consider the rank $2$ semistandard $(3,2)$-parking function in \cref{fig:std-unique} (left). We have that $\Amap(\std(\D, \varphi)) = [1,2,3]$, the identity permutation which does not have inversions and thus has codinv $0$. Then $\dinv(D, \varphi) = \frac{(2-1)(3-1)}{2} - 0 = 1$.  By comparison, the affine permutations corresponding to the two rightmost parking functions in \cref{fig:std-unique} are $[1,3,2]$ and $[2,1,3]$ both of which have inversions. 
\end{example}

%\Nicolle{ add example of dinv computation}

While we do not expand upon this here, it is important to note that the affine spaces in an affine paving of a parabolic affine Springer fiber are indexed by the set $\SSPF^r(m,n)$ and the co-dinv statistic is the dimension of the corresponding affine space. Each parabolic affine Springer fiber admits a surjection from the affine Springer fiber in the full affine flag variety, and the latter variety has an affine paving indexed by parking functions, see \cite{GMV}. The standardization procedure is related to properties of this surjection, see \cite{GSV-2022} for details. 

\begin{definition}\label{def:higher qt}
Let $m,n,r \in \Z_{>0}$ with $\gcd(m,n)=1$. The \newword{rank $r$ rational $(q,t)$-Catalan polynomial} is the polynomial in $\Q[q,t][x_1,\dots, x_r]$ given by
\[C^{(r)}_{(m,n)}(x_1, \dots, x_r;q,t):= \sum_{(\D, \varphi) \in \SSPF^r(m,n)} q^{\area(\D)}t^{\dinv(\D,\varphi)}x^{\wt(\varphi)}.\]
\end{definition}

\begin{example}\label{ex:computation catalan}
It can be shown that $C^{(1)}_{(m,n)}(x_1;q,t) = x_1^mC_{(m,n)}(q,t)$, where $C_{(m,n)}(q,t)$ is the usual $(q,t)$-Catalan polynomial. In this sense, \cref{def:higher qt} generalizes that of $(q,t)$-Catalan polynomials.
As seen in  \cref{fig:rank 2 32}, for $r = 2$, $m = 3$, $n=2$ we have
\[
C^{(2)}_{(3,2)} = (x_1^3 + x_2^3)(q+t) + (x_1^2x_2 + x_1x_2^2)(q+1+t).
\]
\end{example}


\begin{figure}[ht]
\center
\begin{tikzpicture}[scale=0.7]
\begin{scope}[shift={(0,4.5)}] %111
\pgfmathtruncatemacro{\m}{3}
  \pgfmathtruncatemacro{\n}{2}
%%%%%%% DONT TOUCH THIS PART
 \pgfmathtruncatemacro{\mi}{\m-1} %\mi=m-1 
    \pgfmathtruncatemacro{\ni}{\n-1} %\ni=n-1
    %
 % \foreach \x in {0,...,\n} 
    %\foreach \y in {0,...,\m}     
       %{\pgfmathtruncatemacro{\label}{ - \m * \x - \n *  \y +\m*\n} 
       %\node  at (\x-.2,\y-.2) [blue!60!white, scale=.7]{\label};} 
%generates the grid and anderson labels
\draw[dotted] (0,\m)--(\n,0); %draws the diagonal from (0,m) -- (n,0)
  \foreach \x in {0,...,\n} 
    \foreach \y  [count=\yi] in {0,...,\mi}    
      \draw (\x,\y)--(\x,\yi);
        \foreach \y in {0,...,\m}     
       \foreach \x  [count=\xi] in {0,...,\ni}          
         \draw (\x,\y)--(\xi,\y);
 %% Draw dyck path by hand below
 \draw[line width = 2pt,red] (0,3)--(0,0)--(2,0);
 % label the parking function, only modify the x coodinates ! 
\node at (0.5,2.5)[green!20!black]{1};
\node at (0.5,1.5) [green!20!black]{1};
\node at (0.5,0.5)[green!20!black] {1};
%%
\node at (1,-0.5) {$x_1^3q$};
\end{scope}
\begin{scope}[shift={(4,4.5)}] %212
\pgfmathtruncatemacro{\m}{3}
  \pgfmathtruncatemacro{\n}{2}
%%%%%%% DONT TOUCH THIS PART
 \pgfmathtruncatemacro{\mi}{\m-1} %\mi=m-1 
    \pgfmathtruncatemacro{\ni}{\n-1} %\ni=n-1
    %
 % \foreach \x in {0,...,\n} 
    %\foreach \y in {0,...,\m}     
       %{\pgfmathtruncatemacro{\label}{ - \m * \x - \n *  \y +\m*\n} 
       %\node  at (\x-.2,\y-.2) [blue!60!white, scale=.7]{\label};} 
%generates the grid and anderson labels
\draw[dotted] (0,\m)--(\n,0); %draws the diagonal from (0,m) -- (n,0)
  \foreach \x in {0,...,\n} 
    \foreach \y  [count=\yi] in {0,...,\mi}    
      \draw (\x,\y)--(\x,\yi);
        \foreach \y in {0,...,\m}     
       \foreach \x  [count=\xi] in {0,...,\ni}          
         \draw (\x,\y)--(\xi,\y);
 %% Draw dyck path by hand below
 \draw[line width = 2pt,red] (0,3)--(0,1)--(1,1)--(1,0)--(2,0);
 % label the parking function, only modify the x coodinates ! 
\node at (0.5,2.5)[green!20!black]{1};
\node at (0.5,1.5) [green!20!black]{1};
\node at (1.5,0.5)[green!20!black] {1};
%%
\node at (1,-0.5) {$x_1^3t$};
\end{scope}
\begin{scope}[shift={(0,0)}] %321
\pgfmathtruncatemacro{\m}{3}
  \pgfmathtruncatemacro{\n}{2}
%%%%%%% DONT TOUCH THIS PART
 \pgfmathtruncatemacro{\mi}{\m-1} %\mi=m-1 
    \pgfmathtruncatemacro{\ni}{\n-1} %\ni=n-1
    %
 % \foreach \x in {0,...,\n} 
    %\foreach \y in {0,...,\m}     
       %{\pgfmathtruncatemacro{\label}{ - \m * \x - \n *  \y +\m*\n} 
       %\node  at (\x-.2,\y-.2) [blue!60!white, scale=.7]{\label};} 
%generates the grid and anderson labels
\draw[dotted] (0,\m)--(\n,0); %draws the diagonal from (0,m) -- (n,0)
  \foreach \x in {0,...,\n} 
    \foreach \y  [count=\yi] in {0,...,\mi}    
      \draw (\x,\y)--(\x,\yi);
        \foreach \y in {0,...,\m}     
       \foreach \x  [count=\xi] in {0,...,\ni}          
         \draw (\x,\y)--(\xi,\y);
 %% Draw dyck path by hand below
 \draw[line width = 2pt,red] (0,3)--(0,0)--(2,0);
 % label the parking function, only modify the x coodinates ! 
\node at (0.5,2.5)[green!20!black]{2};
\node at (0.5,1.5) [green!20!black]{2};
\node at (0.5,0.5)[green!20!black] {2};
%%
\node at (1,-0.5) {$x_2^3q$};
\end{scope}
\begin{scope}[shift={(4,0)}] %422
\pgfmathtruncatemacro{\m}{3}
  \pgfmathtruncatemacro{\n}{2}
%%%%%%% DONT TOUCH THIS PART
 \pgfmathtruncatemacro{\mi}{\m-1} %\mi=m-1 
    \pgfmathtruncatemacro{\ni}{\n-1} %\ni=n-1
    %
 % \foreach \x in {0,...,\n} 
    %\foreach \y in {0,...,\m}     
       %{\pgfmathtruncatemacro{\label}{ - \m * \x - \n *  \y +\m*\n} 
       %\node  at (\x-.2,\y-.2) [blue!60!white, scale=.7]{\label};} 
%generates the grid and anderson labels
\draw[dotted] (0,\m)--(\n,0); %draws the diagonal from (0,m) -- (n,0)
  \foreach \x in {0,...,\n} 
    \foreach \y  [count=\yi] in {0,...,\mi}    
      \draw (\x,\y)--(\x,\yi);
        \foreach \y in {0,...,\m}     
       \foreach \x  [count=\xi] in {0,...,\ni}          
         \draw (\x,\y)--(\xi,\y);
 %% Draw dyck path by hand below
 \draw[line width = 2pt,red] (0,3)--(0,1)--(1,1)--(1,0)--(2,0);
 % label the parking function, only modify the x coodinates ! 
\node at (0.5,2.5)[green!20!black]{2};
\node at (0.5,1.5) [green!20!black]{2};
\node at (1.5,0.5)[green!20!black] {2};
%%
\node at (1,-0.5) {$x_2^3t$};
\end{scope}
\begin{scope}[shift={(8,4.5)}] %513
\pgfmathtruncatemacro{\m}{3}
  \pgfmathtruncatemacro{\n}{2}
%%%%%%% DONT TOUCH THIS PART
 \pgfmathtruncatemacro{\mi}{\m-1} %\mi=m-1 
    \pgfmathtruncatemacro{\ni}{\n-1} %\ni=n-1
    %
 % \foreach \x in {0,...,\n} 
    %\foreach \y in {0,...,\m}     
       %{\pgfmathtruncatemacro{\label}{ - \m * \x - \n *  \y +\m*\n} 
       %\node  at (\x-.2,\y-.2) [blue!60!white, scale=.7]{\label};} 
%generates the grid and anderson labels
\draw[dotted] (0,\m)--(\n,0); %draws the diagonal from (0,m) -- (n,0)
  \foreach \x in {0,...,\n} 
    \foreach \y  [count=\yi] in {0,...,\mi}    
      \draw (\x,\y)--(\x,\yi);
        \foreach \y in {0,...,\m}     
       \foreach \x  [count=\xi] in {0,...,\ni}          
         \draw (\x,\y)--(\xi,\y);
 %% Draw dyck path by hand below
 \draw[line width = 2pt,red] (0,3)--(0,0)--(2,0);
 % label the parking function, only modify the x coodinates ! 
\node at (0.5,2.5)[green!20!black]{1};
\node at (0.5,1.5) [green!20!black]{1};
\node at (0.5,0.5)[green!20!black] {2};
%%
\node at (1,-0.5) {$x_1^2x_2q$};
\end{scope}
\begin{scope}[shift={(12,4.5)}] %614
\pgfmathtruncatemacro{\m}{3}
  \pgfmathtruncatemacro{\n}{2}
%%%%%%% DONT TOUCH THIS PART
 \pgfmathtruncatemacro{\mi}{\m-1} %\mi=m-1 
    \pgfmathtruncatemacro{\ni}{\n-1} %\ni=n-1
    %
 % \foreach \x in {0,...,\n} 
    %\foreach \y in {0,...,\m}     
       %{\pgfmathtruncatemacro{\label}{ - \m * \x - \n *  \y +\m*\n} 
       %\node  at (\x-.2,\y-.2) [blue!60!white, scale=.7]{\label};} 
%generates the grid and anderson labels
\draw[dotted] (0,\m)--(\n,0); %draws the diagonal from (0,m) -- (n,0)
  \foreach \x in {0,...,\n} 
    \foreach \y  [count=\yi] in {0,...,\mi}    
      \draw (\x,\y)--(\x,\yi);
        \foreach \y in {0,...,\m}     
       \foreach \x  [count=\xi] in {0,...,\ni}          
         \draw (\x,\y)--(\xi,\y);
 %% Draw dyck path by hand below
 \draw[line width = 2pt,red] (0,3)--(0,1)--(1,1)--(1,0)--(2,0);
 % label the parking function, only modify the x coodinates ! 
\node at (0.5,2.5)[green!20!black]{1};
\node at (0.5,1.5) [green!20!black]{1};
\node at (1.5,0.5)[green!20!black] {2};
%%
\node at (1,-0.5) {$x_1^2x_2$};
\end{scope}
\begin{scope}[shift={(16,4.5)}] %715
\pgfmathtruncatemacro{\m}{3}
  \pgfmathtruncatemacro{\n}{2}
%%%%%%% DONT TOUCH THIS PART
 \pgfmathtruncatemacro{\mi}{\m-1} %\mi=m-1 
    \pgfmathtruncatemacro{\ni}{\n-1} %\ni=n-1
    %
 % \foreach \x in {0,...,\n} 
    %\foreach \y in {0,...,\m}     
       %{\pgfmathtruncatemacro{\label}{ - \m * \x - \n *  \y +\m*\n} 
       %\node  at (\x-.2,\y-.2) [blue!60!white, scale=.7]{\label};} 
%generates the grid and anderson labels
\draw[dotted] (0,\m)--(\n,0); %draws the diagonal from (0,m) -- (n,0)
  \foreach \x in {0,...,\n} 
    \foreach \y  [count=\yi] in {0,...,\mi}    
      \draw (\x,\y)--(\x,\yi);
        \foreach \y in {0,...,\m}     
       \foreach \x  [count=\xi] in {0,...,\ni}          
         \draw (\x,\y)--(\xi,\y);
 %% Draw dyck path by hand below
 \draw[line width = 2pt,red] (0,3)--(0,1)--(1,1)--(1,0)--(2,0);
 % label the parking function, only modify the x coodinates ! 
\node at (0.5,2.5)[green!20!black]{1};
\node at (0.5,1.5) [green!20!black]{2};
\node at (1.5,0.5)[green!20!black] {1};
%%
\node at (1,-0.5) {$x_1^2x_2t$};
\end{scope}
\begin{scope}[shift={(8,0)}] %823
\pgfmathtruncatemacro{\m}{3}
  \pgfmathtruncatemacro{\n}{2}
%%%%%%% DONT TOUCH THIS PART
 \pgfmathtruncatemacro{\mi}{\m-1} %\mi=m-1 
    \pgfmathtruncatemacro{\ni}{\n-1} %\ni=n-1
    %
 % \foreach \x in {0,...,\n} 
    %\foreach \y in {0,...,\m}     
       %{\pgfmathtruncatemacro{\label}{ - \m * \x - \n *  \y +\m*\n} 
       %\node  at (\x-.2,\y-.2) [blue!60!white, scale=.7]{\label};} 
%generates the grid and anderson labels
\draw[dotted] (0,\m)--(\n,0); %draws the diagonal from (0,m) -- (n,0)
  \foreach \x in {0,...,\n} 
    \foreach \y  [count=\yi] in {0,...,\mi}    
      \draw (\x,\y)--(\x,\yi);
        \foreach \y in {0,...,\m}     
       \foreach \x  [count=\xi] in {0,...,\ni}          
         \draw (\x,\y)--(\xi,\y);
 %% Draw dyck path by hand below
 \draw[line width = 2pt,red] (0,3)--(0,0)--(2,0);
 % label the parking function, only modify the x coodinates ! 
\node at (0.5,2.5)[green!20!black]{1};
\node at (0.5,1.5) [green!20!black]{2};
\node at (0.5,0.5)[green!20!black] {2};
%%
\node at (1,-0.5) {$x_1x_2^2q$};
\end{scope}
\begin{scope}[shift={(12,0)}] %924
\pgfmathtruncatemacro{\m}{3}
  \pgfmathtruncatemacro{\n}{2}
%%%%%%% DONT TOUCH THIS PART
 \pgfmathtruncatemacro{\mi}{\m-1} %\mi=m-1 
    \pgfmathtruncatemacro{\ni}{\n-1} %\ni=n-1
    %
 % \foreach \x in {0,...,\n} 
    %\foreach \y in {0,...,\m}     
       %{\pgfmathtruncatemacro{\label}{ - \m * \x - \n *  \y +\m*\n} 
       %\node  at (\x-.2,\y-.2) [blue!60!white, scale=.7]{\label};} 
%generates the grid and anderson labels
\draw[dotted] (0,\m)--(\n,0); %draws the diagonal from (0,m) -- (n,0)
  \foreach \x in {0,...,\n} 
    \foreach \y  [count=\yi] in {0,...,\mi}    
      \draw (\x,\y)--(\x,\yi);
        \foreach \y in {0,...,\m}     
       \foreach \x  [count=\xi] in {0,...,\ni}          
         \draw (\x,\y)--(\xi,\y);
 %% Draw dyck path by hand below
 \draw[line width = 2pt,red] (0,3)--(0,1)--(1,1)--(1,0)--(2,0);
 % label the parking function, only modify the x coodinates ! 
\node at (0.5,2.5)[green!20!black]{2};
\node at (0.5,1.5) [green!20!black]{2};
\node at (1.5,0.5)[green!20!black] {1};
%%
\node at (1,-0.5) {$x_1x_2^2$};
\end{scope}
\begin{scope}[shift={(16,0)}] %1025
\pgfmathtruncatemacro{\m}{3}
  \pgfmathtruncatemacro{\n}{2}
%%%%%%% DONT TOUCH THIS PART
 \pgfmathtruncatemacro{\mi}{\m-1} %\mi=m-1 
    \pgfmathtruncatemacro{\ni}{\n-1} %\ni=n-1
    %
 % \foreach \x in {0,...,\n} 
    %\foreach \y in {0,...,\m}     
       %{\pgfmathtruncatemacro{\label}{ - \m * \x - \n *  \y +\m*\n} 
       %\node  at (\x-.2,\y-.2) [blue!60!white, scale=.7]{\label};} 
%generates the grid and anderson labels
\draw[dotted] (0,\m)--(\n,0); %draws the diagonal from (0,m) -- (n,0)
  \foreach \x in {0,...,\n} 
    \foreach \y  [count=\yi] in {0,...,\mi}    
      \draw (\x,\y)--(\x,\yi);
        \foreach \y in {0,...,\m}     
       \foreach \x  [count=\xi] in {0,...,\ni}          
         \draw (\x,\y)--(\xi,\y);
 %% Draw dyck path by hand below
 \draw[line width = 2pt,red] (0,3)--(0,1)--(1,1)--(1,0)--(2,0);
 % label the parking function, only modify the x coodinates ! 
\node at (0.5,2.5)[green!20!black]{1};
\node at (0.5,1.5) [green!20!black]{2};
\node at (1.5,0.5)[green!20!black] {2};
%%
\node at (1,-0.5) {$x_1x_2^2t$};
\end{scope}
\end{tikzpicture}
%%%%\input{figure3-old}   %%% un-comment out if want to see old fig
\caption{All the semistandard $(3,2)$-parking functions of rank $2$ and
their contribution to $C^{(2)}_{(3,2)}(x_1,x_2;q,t)$.}\label{fig:rank 2 32}
\end{figure}
%\Nicolle{add example of higher rank catalan}


Using the standardization procedure above it can be shown that the
restriction to $r$ variables of Gessel's quasisymmetric function
$Q_{\Des(\sigma^{-1})}(X)\vert_{x_1,\dots,x_r}$
is naturally indexed by the rank $r$ semistandard parking functions
$(\D,\varphi)$ that standardize to $(\D,\alpha)$,
where $\Amap(\D,\alpha) = \sigma$
and $\Des(\sigma^{-1})$ is the descent set of $\sigma^{-1}$.
Namely, \begin{equation}
\label{eq:trun-Gessel}
Q_{\Des(\sigma^{-1})}(X)\vert_{x_1,\dots,x_r} = \sum_{(\D,\varphi) \in \std^{-1}(\D,\alpha)} x^{\wt(\D,\varphi)}.
\end{equation}
Combining  \eqref{eq:Hikita} and  \eqref{eq:trun-Gessel} we obtain the following theorem. 
\begin{theorem}\label{thm:Cat-Hik}
Given any $m,n,r \in \Z_{>0}$ with $\gcd(m,n)=1$,
the truncation of the
Hikita polynomial to $r$ variables is given by the rank-$r$ rational
$(q,t)$-Catalan polynomial, \[\Hik_{(m,n)}(X;q,t) \vert_{x_1,\dots,x_r}
= C^{(r)}_{(m,n)}(x_1, \dots, x_r;q,t).\]
Thus, $C^{(r)}_{(m,n)}(x_1, \dots, x_r;q,t)$ is a $(q,t)$-symmetric, Schur positive polynomial, and hence also symmetric in $x_1, \dots, x_r$. 
\end{theorem}

\begin{example}
Let us compute the Hikita polynomial $\Hik_{(3,2)}(X;q,t)$. There are
four $(3,2)$-parking functions $(\D, \varphi)$,  which are listed together
with their associated affine permutations $\Amap(\D, \varphi)$ in
\cref{fig:parking}. From here, we compute the Schur expansion:
\[
\Hik_{(3,2)}(X;q,t) =  qQ_{\{3\}} + Q_{\{2\}} + tQ_{\emptyset} +
Q_{\{1\}}   %%%= (q+t)h_{(3)} + 2m_{(1^3)} + m_{(2,1)}
  = (q+t)s_{(3)} + s_{(2,1)}.
 \]
%\Monica{i did schur expansion instead of mixing h and m which seems weird; added phrase Schur expansion as well, which should be removed if you rever t back to the h m expansion}
It is straightforward to verify that the truncation to $2$-variables of this polynomial coincides with the computation of $C^{(2)}_{(3,2)}(x_1, x_2;q,t)$ in \cref{ex:computation catalan}.
\end{example}

\begin{figure}[ht]
\center
\begin{tikzpicture}[scale=1]
\pgfmathtruncatemacro{\m}{3}
  \pgfmathtruncatemacro{\n}{2}
%%%%%%% DONT TOUCH THIS PART
 \pgfmathtruncatemacro{\mi}{\m-1} %\mi=m-1 
    \pgfmathtruncatemacro{\ni}{\n-1} %\ni=n-1
    %
  \foreach \x in {0,...,\n} 
    \foreach \y in {0,...,\m}     
       {\pgfmathtruncatemacro{\label}{ - \m * \x - \n *  \y +\m*\n} 
       \node  at (\x-.2,\y-.2) [blue!60!white, scale=.7]{\label};} 
%generates the grid and anderson labels
\draw[dotted] (0,\m)--(\n,0); %draws the diagonal from (0,m) -- (n,0)
  \foreach \x in {0,...,\n} 
    \foreach \y  [count=\yi] in {0,...,\mi}    
      \draw (\x,\y)--(\x,\yi);
        \foreach \y in {0,...,\m}     
       \foreach \x  [count=\xi] in {0,...,\ni}          
         \draw (\x,\y)--(\xi,\y);
 %% Draw dyck path by hand below
 \draw[line width = 2pt,red] (0,3)--(0,0)--(2,0);
 % label the parking function, only modify the x coodinates ! 
\node at (0.5,2.5)[green!20!black]{1};
\node at (0.5,1.5) [green!20!black]{2};
\node at (0.5,0.5)[green!20!black] {3};
%%
\node at (1,-1) {$[0,2,4]$};
\end{tikzpicture}
\qquad\qquad
\begin{tikzpicture}[scale=1]
\pgfmathtruncatemacro{\m}{3}
  \pgfmathtruncatemacro{\n}{2}
%%%%%%% DONT TOUCH THIS PART
 \pgfmathtruncatemacro{\mi}{\m-1} %\mi=m-1 
    \pgfmathtruncatemacro{\ni}{\n-1} %\ni=n-1
    %
  \foreach \x in {0,...,\n} 
    \foreach \y in {0,...,\m}     
       {\pgfmathtruncatemacro{\label}{ - \m * \x - \n *  \y +\m*\n} 
       \node  at (\x-.2,\y-.2) [blue!60!white, scale=.7]{\label};} 
%generates the grid and anderson labels
\draw[dotted] (0,\m)--(\n,0); %draws the diagonal from (0,m) -- (n,0)
  \foreach \x in {0,...,\n} 
    \foreach \y  [count=\yi] in {0,...,\mi}    
      \draw (\x,\y)--(\x,\yi);
        \foreach \y in {0,...,\m}     
       \foreach \x  [count=\xi] in {0,...,\ni}          
         \draw (\x,\y)--(\xi,\y);
 %% Draw dyck path by hand below
 \draw[line width = 2pt,red] (0,3)--(0,1)--(1,1)--(1,0)--(2,0);
 % label the parking function, only modify the x coodinates ! 
\node at (0.5,2.5)[green!20!black]{1};
\node at (0.5,1.5) [green!20!black]{2};
\node at (1.5,0.5)[green!20!black] {3};
%%
\node at (1,-1) {$[1,3,2]$};
\end{tikzpicture}
\qquad\qquad
\begin{tikzpicture}[scale=1]
\pgfmathtruncatemacro{\m}{3}
  \pgfmathtruncatemacro{\n}{2}
%%%%%%% DONT TOUCH THIS PART
 \pgfmathtruncatemacro{\mi}{\m-1} %\mi=m-1 
    \pgfmathtruncatemacro{\ni}{\n-1} %\ni=n-1
    %
  \foreach \x in {0,...,\n} 
    \foreach \y in {0,...,\m}     
       {\pgfmathtruncatemacro{\label}{ - \m * \x - \n *  \y +\m*\n} 
       \node  at (\x-.2,\y-.2) [blue!60!white, scale=.7]{\label};} 
%generates the grid and anderson labels
\draw[dotted] (0,\m)--(\n,0); %draws the diagonal from (0,m) -- (n,0)
  \foreach \x in {0,...,\n} 
    \foreach \y  [count=\yi] in {0,...,\mi}    
      \draw (\x,\y)--(\x,\yi);
        \foreach \y in {0,...,\m}     
       \foreach \x  [count=\xi] in {0,...,\ni}          
         \draw (\x,\y)--(\xi,\y);
 %% Draw dyck path by hand below
 \draw[line width = 2pt,red] (0,3)--(0,1)--(1,1)--(1,0)--(2,0);
 % label the parking function, only modify the x coodinates ! 
\node at (0.5,2.5)[green!20!black]{1};
\node at (0.5,1.5) [green!20!black]{3};
\node at (1.5,0.5)[green!20!black] {2};
%%
\node at (1,-1) {$[1,2,3]$};
\end{tikzpicture}
\qquad\qquad
\begin{tikzpicture}[scale=1]
\pgfmathtruncatemacro{\m}{3}
  \pgfmathtruncatemacro{\n}{2}
%%%%%%% DONT TOUCH THIS PART
 \pgfmathtruncatemacro{\mi}{\m-1} %\mi=m-1 
    \pgfmathtruncatemacro{\ni}{\n-1} %\ni=n-1
    %
  \foreach \x in {0,...,\n} 
    \foreach \y in {0,...,\m}     
       {\pgfmathtruncatemacro{\label}{ - \m * \x - \n *  \y +\m*\n} 
       \node  at (\x-.2,\y-.2) [blue!60!white, scale=.7]{\label};} 
%generates the grid and anderson labels
\draw[dotted] (0,\m)--(\n,0); %draws the diagonal from (0,m) -- (n,0)
  \foreach \x in {0,...,\n} 
    \foreach \y  [count=\yi] in {0,...,\mi}    
      \draw (\x,\y)--(\x,\yi);
        \foreach \y in {0,...,\m}     
       \foreach \x  [count=\xi] in {0,...,\ni}          
         \draw (\x,\y)--(\xi,\y);
 %% Draw dyck path by hand below
 \draw[line width = 2pt,red] (0,3)--(0,1)--(1,1)--(1,0)--(2,0);
 % label the parking function, only modify the x coodinates ! 
\node at (0.5,2.5)[green!20!black]{2};
\node at (0.5,1.5) [green!20!black]{3};
\node at (1.5,0.5)[green!20!black] {1};
%%
\node at (1,-1) {$[2,1,3]$};
\end{tikzpicture}
\caption{The $(3,2)$-parking functions and their associated affine permutations under the bijection $\Amap$, which can easily be read from
 $\color{blue} \gamma (x,y)$ for each lattice point $\color{red}(x,y)$
 on $\D$.
%the values of $\color{blue} \gamma$ in each lattice point on $\D$.
}\label{fig:parking}
\end{figure}
%\Monica{should we also mention in the Thus that is Schur positive?}
\section{Spherical DAHA and a Finite Shuffle Theorem}

%\cref{thm:Cat-Hik} yields a direct relation between the higher rank Catalan polynomials and the Cherednik's double affine Hecke algebra (DAHA). To do so we recall some important definitions. 

The \newword{elliptic Hall algebra} $\EHA$ is the $\C(q,t)$-algebra
consisting of an infinite family of generators $\{ P_{(m,n)} \; | \;
(m,n) \in \Z^{2}_{\geq 0}\setminus (0,0)\}$ modulo some relations, see e.g. \cite{SV}. %It is a theorem of
%\cite{??} that $\EHA$ can be generated by only $P_{(0,1)}$ and
%$P_{(1,0)}$.
%\Monica{out of Schiffmann-Vasserot \cite{SV}, Burban-Schiffmann (see
%their 6.1) \cite{BurbanSchiffmann}, and  Morton-Samuelson
%\cite{MortonSamuelson}, i'd say ?? = it's there in SV depending on which
%relations one means. they have $t_x$
%which may not be these $P_{mn}$ . by erasing this comment, the cites
%will disappear }
Although $\EHA$ originally arose as the Hall algebra of
the category of coherent sheaves on an elliptic curve, it has since had
various topological and combinatorial incarnations and played a central
role in connecting $q,t$-combinatorics and the Shuffle Theorems to many
recent results in the theory of homological and geometric invariants
for knots and links. 

There is a well-known geometric action of $\EHA$ on the ring of symmetric functions $\Lambda_{q,t}$ whose fixed points correspond to the \emph{modified Macdonald polynomials} $\widetilde{H}_\lambda(X;q,t)$ \cite{SV}. Denote this representation by $V^{\geom}$ and by $P_{(m,n)}^{\;\infty}(f)$ the image of $f \in \Lambda_{q,t}$ under this action. The \emph{Rational Shuffle Theorem}, conjectured by Gorsky and Negu\c{t} \cite{GN} and proved by Mellit \cite{Mellit}, states the following. 
\begin{theorem}[\cite{Mellit}] \label{thm:shuffle}
For coprime $m,n \in \Z_{>0}$, \,
$P_{(m,n)}^{\;\infty} (1) = \Hik_{(m,n)}(X;q,t)$.
\end{theorem}

%\subsection{Spherical DAHA and a Finite Shuffle Theorem}

The \newword{spherical DAHA} $\mathbb{SH}(r)$ is the spherical subalgebra of the double affine Hecke algebra of type $\mathfrak{gl}(r)$. We consider the subalgebra $\DAHA{r}$ of $\mathbb{SH}(r)$ generated by elements $P^{(r)}_{(m,n)}$
with $(m,n)\in \Z^2_{\geq 0}\setminus (0,0)$. Schiffmann-Vasserot show that under the natural map $\DAHA{r} \to \DAHA{r-1}$ sending $P^{(r)}_{(m,n)} \mapsto P^{(r-1)}_{(m,n)}$, the algebra $\EHA$ arises as an inverse limit of the spherical DAHAs. Namely, 
\begin{equation}\label{eq:EHA-invlim}
\EHA \cong \varprojlim_{r}\DAHA{r}
\end{equation}
under which 
\[P_{(m,n)} = \varprojlim_{r} P_{(m,n)}^{(r)}.\]
The algebra $\DAHA{r}$ comes equipped with a faithful polynomial representation on $\Q(q,t)[x_1\dots, x_r]^{S_r}$, denoted $Pol(r)$, that is compatible with \eqref{eq:EHA-invlim} above and the inverse limit $\Lambda_{q,t} \cong \varprojlim_{r} \Q(q,t)[x_1\dots, x_r]^{S_r}$. Thus, there is an induced polynomial representation $V^{\alg}$ of $\EHA$ on $\Lambda_{q,t}$. This representation is nontrivially isomorphic to the geometric one of Schiffmann-Vasserot. 

\begin{proposition}[\cite{SV}]
There is an isomorphism of $\EHA$ representations $\Phi: V^{\alg} \to V^{\geom}$ given by the plethystic substitution that sends $F(X;q,t) \mapsto F \left[ \frac{X}{1-t^{-1}} ; q, t^{-1}\right]$.
\end{proposition}

By combining this proposition with \cref{thm:Cat-Hik,thm:shuffle} we obtain the following finite analogue of the Rational Shuffle Theorem. 

\begin{theorem}\label{thm:finite-shuffle}
Given any $m,n,r \in \Z_{>0}$ with $\gcd(m,n)=1$,
%Given any positive integer $r$ and any pair of coprime integers $(m,n)$,
there exists an action of $\DAHA{r}$ on $\Q(q,t)[x_1\dots, x_r]^{S_r}$ satisfying
\[P_{(m,n)}^{(r)}(1) = C_{(m,n)}^{(r)}(x_1,\dots, x_r ; q,t).\]
In particular, this action is nontrivially isomorphic to the polynomial representation $Pol(r)$. 
\end{theorem}

This action is compatible with the geometric action of $\EHA$ on $\Lambda_{q,t}$ under all the inverse limits above, with $P_{(m,n)}^{\; \infty} = \varprojlim_{r} P_{(m,n)}^{(r)}$ as operators, so that as $r \to \infty$ it recovers the Rational Shuffle Theorem.

Given the complicated nature of plethystic substitution, finding the explicit action of $\DAHA{r}$ on $\Q(q,t)[x_1\dots, x_r]^{S_r}$ in \cref{thm:finite-shuffle} is quite difficult. In particular, even describing the action for $r=1$ explicitly is unclear since although $\DAHA{1}$ is straightforward, the rational Catalan numbers $C_{(m,n)}(q,t)$ are not.
Nonetheless, it would be very interesting to find an explicit description of this action.
%% and its relationship to the usual polynomial representation $Pol(r)$.


\section{The Non-Coprime Case and a Bizley-Type Formula}

Naturally, one asks how much of these constructions can be extended to the non-coprime case. Unfortunately it is neither easy nor obvious how to proceed. A starting point, however, is the computation of $C_{(m,n)}^{(r)} = |\SSPFr|$ in the
%noncoprime case
case $m, n$ are not coprime.
 %of the higher rank Catalan numbers $C_{(m,n)}^{(r)}$ defined in \eqref{eq:mn-Catalan-numbers}. 
Below we give a Bizley-type formula for these numbers.

\begin{theorem}
Let $m,n,r \in \Z_{>0}$ with $\gcd(m,n)=1$.
% Let $m, n$ be coprime positive integers, and $r > 0$.
There is an equality of formal power series:
\[
1 + \sum_{k = 1}^{\infty}C^{(r)}_{(mk,nk)}z^{k} = \exp\left(\sum_{k = 1}^{\infty}\frac{1}{nk}\binom{nkr+mk-1}{mk}z^{k} \right).
\]
\end{theorem}

One can apply similar techniques to obtain a Bizley-type formula for
the number of parking functions. %\Monica{just parking or SSPF too?}
 This is, however, much more elegantly
expressed by Aval and Bergeron in \cite{AvalBergeron}, who adapt
Bizley's proof incorporating symmetric function arguments to find the
Frobenius characteristic of the $S_{mk}$-representation
$\C\!\PF(mk,nk)$. One can adapt the arguments in this paper to show
that \[
C^{(r)}_{(mk,nk)} = \operatorname{Frob}(\C\!\PF(mk, nk))|_{x_{1} = \cdots = x_{mk} = 1, x_{mk+1} = \cdots = 0}.
\]




%\acknowledgements{We thank ... }

%% MV - i think ok to just say Simons, but in case of more details....
%Program Mathematics and Physical Sciences---Collaboration Grants for Mathematicians 
%Project Title Hecke algebras in combinatorial representation theory 707426

%% 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}



\begin{tikzpicture}
% pick an m and n 
 \pgfmathtruncatemacro{\m}{5}
  \pgfmathtruncatemacro{\n}{7}
%%%%%%% DONT TOUCH THIS PART
 \pgfmathtruncatemacro{\mi}{\m-1} %\mi=m-1 
    \pgfmathtruncatemacro{\ni}{\n-1} %\ni=n-1
    %
  \foreach \x in {0,...,\n} 
    \foreach \y in {0,...,\m}     
       {\pgfmathtruncatemacro{\label}{ - \m * \x - \n *  \y +\m*\n} 
       \node  at (\x-.2,\y-.2) [blue!60!white, scale=.7]{\label};} 
%generates the grid and anderson labels
\draw[dotted] (0,\m)--(\n,0); %draws the diagonal from (0,m) -- (n,0)
  \foreach \x in {0,...,\n} 
    \foreach \y  [count=\yi] in {0,...,\mi}    
      \draw (\x,\y)--(\x,\yi);
        \foreach \y in {0,...,\m}     
       \foreach \x  [count=\xi] in {0,...,\ni}          
         \draw (\x,\y)--(\xi,\y);
 %% Draw dyck path by hand below
\end{tikzpicture}


\begin{figure}[ht]
\center{\begin{tikzpicture}
% pick an m and n 
 \pgfmathtruncatemacro{\m}{7}
  \pgfmathtruncatemacro{\n}{5}
%%%%%%% DONT TOUCH THIS PART
 \pgfmathtruncatemacro{\mi}{\m-1} %\mi=m-1 
    \pgfmathtruncatemacro{\ni}{\n-1} %\ni=n-1
    %
  \foreach \x in {0,...,\n} % n=5
    \foreach \y in {0,...,\m}     % m=7  
       {\pgfmathtruncatemacro{\label}{ - \m * \x - \n *  \y +\m*\n} % - m * \x - n *  \y +mn
      % \node   (\x\y) at (\x,\y){};
       \node  at (\x-.2,\y-.2) [blue!60!white, scale=.7]{\label};} 
%generates the grid and anderson labels
\draw[dotted] (0,\m)--(\n,0); %draws the diagonal from (0,m) -- (n,0)
  \foreach \x in {0,...,\n} % n=5
    \foreach \y  [count=\yi] in {0,...,\mi}   % m-1=6  
    %\draw (\x\y)--(\x\yi);
      \draw (\x,\y)--(\x,\yi);
        \foreach \y in {0,...,\m}     % m=7  
       \foreach \x  [count=\xi] in {0,...,\ni}   % n-1=4
        %\draw (\x\y)--(\xi\y);
         \draw (\x,\y)--(\xi,\y);
%%%%%%%%%%%%%%%%%%%%%%
 % draw Dyck path
\draw[line width = 2pt,red] (0,7) --(0,5) -- (1,5)--(1,3)--(2,3)--(2,0)--(5,0); 
% label the parking function, only modify the x coodinates ! 
\node at (.5,6.5) [green!20!black]{8};
\node at (.5,5.5) [green!20!black]{10};
\node at (1.5,4.5) [green!20!black]{1};
\node at (1.5,3.5) [green!20!black]{7};
\node at (2.5,2.5)[green!20!black]{2};
\node at (2.5,1.5) [green!20!black]{2};
\node at (2.5,0.5)[green!20!black] {3};
\end{tikzpicture}
$\qquad \qquad$


\begin{figure}[ht]
\center{\begin{tikzpicture}
% pick an m and n 
 \pgfmathtruncatemacro{\m}{7}
  \pgfmathtruncatemacro{\n}{5}
%%%%%%% DONT TOUCH THIS PART
 \pgfmathtruncatemacro{\mi}{\m-1} %\mi=m-1 
    \pgfmathtruncatemacro{\ni}{\n-1} %\ni=n-1
    %
  \foreach \x in {0,...,\n} % n=5
    \foreach \y in {0,...,\m}     % m=7  
       {\pgfmathtruncatemacro{\label}{ - \m * \x - \n *  \y +\m*\n} % - m * \x - n *  \y +mn
      % \node   (\x\y) at (\x,\y){};
       \node  at (\x-.2,\y-.2) [blue!60!white, scale=.7]{\label};} 
%generates the grid and anderson labels
\draw[dotted] (0,\m)--(\n,0); %draws the diagonal from (0,m) -- (n,0)
  \foreach \x in {0,...,\n} % n=5
    \foreach \y  [count=\yi] in {0,...,\mi}   % m-1=6  
    %\draw (\x\y)--(\x\yi);
      \draw (\x,\y)--(\x,\yi);
        \foreach \y in {0,...,\m}     % m=7  
       \foreach \x  [count=\xi] in {0,...,\ni}   % n-1=4
        %\draw (\x\y)--(\xi\y);
         \draw (\x,\y)--(\xi,\y);
%%%%%%%%%%%%%%%%%%%%%%
 % draw Dyck path
\draw[line width = 2pt,red] (0,7) --(0,5) -- (1,5)--(1,3)--(2,3)--(2,0)--(5,0); 
% label the parking function, only modify the x coodinates ! 
\node at (.5,6.5) [green!20!black]{5};
\node at (.5,5.5) [green!20!black]{7};
\node at (1.5,4.5) [green!20!black]{1};
\node at (1.5,3.5) [green!20!black]{6};
\node at (2.5,2.5)[green!20!black]{2};
\node at (2.5,1.5) [green!20!black]{3};
\node at (2.5,0.5)[green!20!black] {4};
\end{tikzpicture}
$\qquad \qquad$
\begin{tikzpicture}
% pick an m and n 
 \pgfmathtruncatemacro{\m}{7}
  \pgfmathtruncatemacro{\n}{5}
%% DONT TOUCH THE REST
 \pgfmathtruncatemacro{\mi}{\m-1} %\mi=m-1 
    \pgfmathtruncatemacro{\ni}{\n-1} %\ni=n-1
    %
  \foreach \x in {0,...,\n} % n=5
    \foreach \y in {0,...,\m}     % m=7  
       {\pgfmathtruncatemacro{\label}{ - \m * \x - \n *  \y +\m*\n} % - m * \x - n *  \y +mn
      % \node   (\x\y) at (\x,\y){};
       \node  at (\x-.2,\y-.2) [blue!60!white, scale=.7]{\label};} 
%generates the grid and anderson labels
\draw[dotted] (0,\m)--(\n,0); %draws the diagonal from (0,m) -- (n,0)
  \foreach \x in {0,...,\n} % n=5
    \foreach \y  [count=\yi] in {0,...,\mi}   % m-1=6  
    %\draw (\x\y)--(\x\yi);
      \draw (\x,\y)--(\x,\yi);
        \foreach \y in {0,...,\m}     % m=7  
       \foreach \x  [count=\xi] in {0,...,\ni}   % n-1=4
        %\draw (\x\y)--(\xi\y);
         \draw (\x,\y)--(\xi,\y);
 % draw Dyck path by hand
\draw[line width = 2pt,red] (0,7) --(0,5) -- (1,5)--(1,3)--(2,3)--(2,0)--(5,0); 
% label the parking function, only modify the x coodinates ! 
\node at (.5,6.5) [green!20!black]{1};
\node at (.5,5.5) [green!20!black]{7};
\node at (1.5,4.5) [green!20!black]{4};
\node at (1.5,3.5) [green!20!black]{5};
\node at (2.5,2.5)[green!20!black]{2};
\node at (2.5,1.5) [green!20!black]{3};
\node at (2.5,0.5)[green!20!black] {6};
\end{tikzpicture}}
\caption{The standardizations of \cref{fig:7-5-PF}} \label{fig:7-5std}
\end{figure}




