
\documentclass[finalversion]{FPSAC2020}
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%% define your title in the usual way
\title[Compact formulas for Macdonald polynomials]{Compact formulas for Macdonald polynomials and quasisymmetric Macdonald polynomials}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author[S.~Corteel, J.~Haglund, O.~Mandelshtam, S.~Mason, and L.~Williams]{Sylvie Corteel\thanks{\href{mailto:corteel@berkeley.edu}{corteel@berkeley.edu}}\addressmark{1}, Jim Haglund \addressmark{2}\thanks{\href{mailto:haglund@math.upenn.edu}{haglund@math.upenn.edu} Partially supported by NSF grant DMS--1600670.}, Olya Mandelshtam\addressmark{4}\thanks{\href{mailto:olya@math.brown.edu}{olya@math.brown.edu} Partially supported by NSF grant DMS--1704874.},\\ Sarah Mason\addressmark{3}\thanks{\href{mailto:masonsk@wfu.edu}{masonsk@wfu.edu}}, \and Lauren Williams\addressmark{5}
\thanks{\href{mailto:williams@math.harvard.edu}{williams@math.harvard.edu} Partially supported by NSF grant DMS--1854512.}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1} Department of Mathematics, UC Berkeley, CA, USA\\ \addressmark{2}
Department of Mathematics, University of Pennsylvania, Philadelphia, PA, USA\\ \addressmark{3} Department of Mathematics, Brown University, Providence, RI, USA
\\ \addressmark{4} Department of Mathematics, Wake Forest University, Winston-Salem, NC, USA\\ \addressmark{5} Department of Mathematics, Harvard
University, Cambridge, MA, USA}

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

\abstract{We present several new and compact formulas for the modified and integral form of the Macdonald polynomials, building on the 
compact ``multiline queue'' formula for Macdonald polynomials 
due to Corteel, Mandelshtam and Williams. 
We also introduce 
a new quasisymmetric analogue of Macdonald polynomials.  These \emph{quasisymmetric Macdonald polynomials} refine 
the (symmetric) Macdonald polynomials and specialize
at $q=t=0$ to the quasisymmetric Schur polynomials defined by Haglund, Luoto, Mason, and Van Willigenburg.}

%% put your French abstract here, or comment this out if you don't have one
\resume{Nous pr\'esentons plusieurs formules nouvelles et compactes pour les poly\-n\^omes de Macdonald modifi\'es et
la forme int\'egrale des polyn\^omes de Macdonald. Nous utilisons des id\'ees venant des formules combinatoires
 pour les polyn\^omes de Macdonald en termes de ``file d'attente \`a plusieurs lignes''
de Corteel, Mandelshtam et Williams. Nous d\'efinissons aussi des polyn\^omes de Macdonald quasisym\'etriques
qui sont un rafinement des polyn\^omes de Macdonald et ont la bonne propri\'et\'e  quand $q=t=0$ d\^etre \'egaux
aux fonctions de Schur quasisym\'etriques d\'efinies par Haglund, Luoto, Mason et Van Willigenburg.
}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{Macdonald polynomials, quasisymmetric polynomials, asymmetric simple exclusion process.}



\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 symmetric 
\emph{Macdonald polynomials} $P_{\lambda}(X; q, t)$ \cite{Macdonald}
are a family of polynomials in 
$X = \{x_1, x_2,\dots \}$ indexed by partitions, whose coefficients depend on two parameters $q$ and $t$.  Macdonald polynomials generalize 
multiple important families of 
polynomials, including Schur polynomials and Hall-Littlewood polynomials.
 They can be defined as the unique monic basis for the ring of symmetric functions that satisfies certain triangularity and orthogonality conditions. The related \emph{nonsymmetric
 Macdonald polynomials} $E_{\mu}(X;q,t)$ \cite{Mac88, Macdonald, Cher1}
 first appeared shortly after the introduction of Macdonald polynomials as a tool to study Macdonald polynomials.
 The $E_{\mu}(X;q, t)$ 
  are indexed by weak compositions and form a basis for the full polynomial ring $\mathbb{Q}[X](q,t)$.
  
 There has been a great deal of work devoted to understanding Macdonald
polynomials from a combinatorial point of view.
Haglund-Haiman-Loehr \cite{HHL04} gave a combinatorial formula for the 
\emph{integral forms} $J_{\lambda}(X; q,t)$, which are scalar multiples
of the classical monic forms $P_{\lambda}(X;q, t)$.  They also gave a 
formula for the 
nonsymmetric Macdonald polynomials $E_{\mu}(X;q,t)$ \cite{HHL08}, and for the
\emph{transformed} or \emph{modified} Macdonald polynomials
$\widetilde{H}_{\lambda}(X;q,t)$, which are obtained from $J_{\lambda}(X;q, t)$ via \emph{plethysm}. Macdonald conjectured and Haiman proved \cite{Hai99}, using the 
geometry of the Hilbert scheme, that the modified Macdonald polynomials 
$\widetilde{H}_{\lambda}(X;q,t)$ 
have a positive Schur expansion whose coefficients are $qt$-Kostka polynomials.  However, it is still an open
problem to give a combinatorial proof of Schur positivity or a 
manifestly positive formula 
for the $qt$-Kostka polynomials.

Recently a beautiful connection has been found between Macdonald polynomials
and 
a model from statistical mechanics called the multispecies \emph{asymmetric simple exclusion process} (ASEP) on a circle. The ASEP is a one-dimensional exactly solvable particle model; Cantini-deGier-Wheeler \cite{CGW} showed that the 
partition function of the multispecies ASEP on a circle
is equal to a Macdonald polynomial
$P_{\lambda}(x_1,\dots,x_n; q, t)$ evaluated at $q=1$ and $x_i=1$ for all $i$.
Building on this result as well as work of Martin \cite{Martin},
 the first, third, and fifth authors recently used \emph{multiline queues}
 to simultaneously compute the stationary probabilities of the multispecies exclusion process, and give compact formulas for the symmetric 
 Macdonald polynomials $P_{\lambda}$ and the nonsymmetric Macdonald
 polynomials  $E_{\lambda}$ \cite{CMW18}, for any partition $\lambda$.  
 These formulas are ``compact'' in that they have fewer terms than the formulas of Haglund-Haiman-Loehr. 


In this paper we use the above ideas to 
continue the search for compact formulas for Macdonald polynomials.  Our first two main results are compact formulas for the modified
Macdonald polynomials 
$\widetilde{H}_{\lambda}(X;q,t)$ 
and the integral forms  $J_{\lambda}(X;q,t)$; these new formulas have far fewer
terms than other known combinatorial formulas.  
Our third main result uses  the connection with the ASEP on a ring 
towards a different application: the introduction of a new family 
of quasisymmetric functions we call \emph{quasisymmetric Macdonald 
polynomials}  $G_{\gamma}(X; q, t)$.  We show that $G_{\gamma}(X; q, t)$
is indeed a quasisymmetric function and we have a
combinatorial formula for the $G_{\gamma}(X; q, t)$, corresponding to ``pieces'' of the compact formula for the $P_{\lambda}$ from \cite{CMW18}. 
The Macdonald polynomial $P_{\lambda}(X; q, t)$ is a sum of these quasisymmetric
Macdonald polynomials, and the quasisymmetric function $G_{\gamma}(X; q, t)$ at $q=t=0$  specializes
to the \emph{quasisymmetric Schur functions} $\text{QS}_{\gamma}(X)$ 
introduced by the second and fourth authors, together with Luoto and van Willigenburg 
\cite{HLMV09}.  The quasisymmetric Schur functions form a basis
for the ring of quasisymmetric functions and until now it has been an 
open question to find a refinement of the Macdonald polynomials 
$P_{\lambda}$ into quasisymmetric pieces which generalize the 
quasisymmetric Schur functions. 

This paper is organized as follows.  In 
\cref{sec:def}, we provide the relevant background. \cref{sec:Hcompact,sec:Jcompact} describe our two compact formulas, and 
\cref{sec:qsym} defines our new quasisymmetric Macdonald polynomials. See the full paper \cite{JLOSS} for the proofs of our results. 




\section{Definitions}\label{sec:def}

We begin by introducing relevant notation and definitions.  In our partition/composition diagrams, given in French notation, the columns are labeled from left to right, and the rows are labeled from bottom to top, so that the notation $(i,r)$ refers to the box (or \emph{cell}) in the $i^{th}$ column from the left and the $r^{th}$ row from the bottom. Given a partition/composition $\alpha$, its diagram $\dg(\alpha)$ is a sequence of columns bottom justified where the $i^{th}$ column has $\alpha_i$ cells.  The cell $(i,r)$ refers to the cell in row $r$ of column $i$. The \emph{leg} of a cell $(i,r)$, denoted $\leg((i,r))$, equals the number of cells in column $i$ above the cell $(i,r)$.  Analogously the \emph{arm} of a cell $(i,r)$, denoted $\arm((i,r))$, equals the number of cells in row $r$ to the right of the cell $(i,r)$ contained in a column whose height doesn't exceed $\alpha_i$, plus the number of cells in row $r-1$ to the left of the cell $(i,r-1)$, contained in a column whose height is smaller than $\alpha_i$.  
See \cref{fig:leg}. 


\begin{figure}
  \centering
\begin{tikzpicture}[scale=.5]
\cell31{$$} \cell32{} \cell33{$$} \cell34{$u$}  \cell37{$a$}\cell39{$a$}
\cell41{} \cell42{} \cell43{$a$}\cell44{}  \cell46{$$} \cell47{} \cell49{$$} %\cell49{}
\cell12{} 
\cell21{} \cell22{} \cell24{$\ell$} \cell27{}
\cell51{$$} \cell52{$$}  \cell53{$$} \cell54{$$}  \cell55{$$}  \cell56{$$}  \cell57{$$}  \cell58{$$}  \cell59{$$} 
\end{tikzpicture} 
\caption{The diagram of the composition $(4,5,3,4,1,2,4,1,3)$  and the cells in the leg and the arm of the cell $u=(4,3)$. Here $\leg(u)=1$ and $\arm(u)=3$.}
\label{fig:leg}
\end{figure}


In \cref{sec:Hcompact}, we only consider diagrams corresponding to partitions. Given a partition $\lambda$, a \emph{filling} $\sigma:\dg(\lambda) \to \Z^+$ is an assignment of positive integers to the cells of $\dg(\lambda)$ and is denoted by $\sigma$.\footnote{Note that $\dg(\lambda)$ corresponds to the Ferrers diagram of $\lambda'$ in French notation.}  
 

For $s\in \dg(\lambda)$, let $\sigma(s)$ denote the integer assigned to $s$, i.e. the integer occupying cell $s$.  The numbers appearing in such a filling are called the \emph{entries}. 
For each filling $\sigma$ of $\dg(\lambda)$ we associate $x$, $q$, and $t$ \emph{weights}.  The \emph{$x$-weight} is defined in a similar fashion to semistandard Young tableaux, namely \[x^{\sigma} = \prod_{s\in \dg(\lambda)} x_{\sigma(s)}.\]

We recall several definitions from~\cite{HHL04}.
Assume that the diagram of a partition $\lambda$ has a \emph{basement}, i.e.
a zero(th) row of size $\lambda_1$ all of whose cells are filled with the entry $\infty$.
Let $u$, $v$, and $r$ be positive integers with $u<v$.
	 Given a (diagram of a) partition $\lambda$ and a filling $\sigma$ of $\lambda$,
	a \emph{triple} consists of the three cells (if they are present in the diagram)
 $(v,r)$, $(u,r)$, and $(u,r-1)$. 
	 Let $a = \sigma(v,r)$, $b = \sigma(u,r)$, and $c=\sigma(u,r-1)$.  	
	We say that the triple  is a \emph{counterclockwise inversion triple} if 
	any of the following holds:
\begin{align*}
a < b \leq c\ {\rm or}\
c < a < b \ {\rm or}\
b \leq c < a.
\end{align*}
For example, in \cref{fig:perm} the entries $(3,3)$, $(1,3)$ and $(1,2)$  form a counterclockwise inversion triple.
We say that the triple  is a \emph{clockwise inversion triple} if 
	any of the following holds:
\begin{align*}
a > b > c\ {\rm or}\
c > a > b \ {\rm or}\
b > c > a.
\end{align*}


Note that since $\sigma(j,0)=\infty$, if $r=1$ then $(v,1)$ and  $(u,1)$ form a (counterclockwise) inversion triple if and only if
$\sigma(u,1)>\sigma(v,1)$. In this case we say that the triple is {\em degenerate}. For example, on \cref{fig:perm} the entries $(3,1)$ and $(6,1)$  form a degenerate inversion triple.
Given a filling $\sigma$, let $\inv(\sigma)$ be the total number of  (counterclockwise) inversion triples including degenerate triples.  Let $\coinv(\sigma)$ be the total number of triples in the filling minus $\inv(\sigma)$.  (In other words, $\coinv(\sigma)$ is the number of triples which are not counterclockwise inversion triples.)

Define the \emph{set of descents} of a filling to be 
\[
\Des(\lambda, \sigma) = \{(u,r)\in \lambda\ :\ \sigma((u,r))>\sigma((u,r-1))\}.
\]   
We define the \emph{major index} $\maj(\sigma)$ to be the sum over the legs of the descents of 
	$(\lambda, \sigma)$:
\[
\maj(\sigma) = \sum_{x \in \Des(\sigma)} (\leg(x)+1).
\]
The tableau $\sigma$ of $\dg(\lambda)$ in \cref{fig:perm} has $\inv(\sigma,\lambda)=22$ and $\maj(\sigma,\lambda)=5$.
Two cells are \emph{attacking} if
their entries are equal and the cells are either in the same row, or they are in adjacent
rows, with the rightmost cell in a row strictly below the other cell. A filling is \emph{non-attacking}
if it does not contain any attacking pairs of cells.

Given a \emph{weak composition}, i.e. a 
vector $\alpha = (\alpha _1, \alpha _2, \ldots , \alpha _n)$ of nonnegative integers, we let
$\inc(\alpha)$ 
and $\dec(\alpha)$ 
be the vectors obtained from $\alpha$ by sorting the parts in 
weakly increasing order, and weakly decreasing order, respectively.
Let $\beta (\alpha)$ be the permutation in $S_n$ of maximal length with the property that
 $\beta$ applied to the vector $\alpha$ yields $\inc(\alpha)$.  Let $\alpha ^{+}$ be the \emph{strong composition} obtained from
 $\alpha$ by removing  the zeros and let $\ell (\alpha)$ be the number of parts of $\alpha ^{+}$.  For example, if $\alpha = (0,2,0,2,1, 3)$ then 
 $\inc(\alpha) = (0,0,1,2,2,3)$, $\dec(\alpha) = (3,2,2,1,0,0)$, $\beta (\alpha) = (3,1,5,4,2,6)$, $\alpha ^{+} = (2,2,1,3)$, and $\ell (\alpha) = 4$.     

\section{Compact formula for modified Macdonald polynomials}\label{sec:Hcompact}


\begin{figure}

  \centering

\begin{tikzpicture}[scale=.5]

\node at (-1.5,.5) {\rm row 5};
\node at (-1.5,-.5) {\rm row 4};
\node at (-1.5,-1.5) {\rm row 3};
\node at (-1.5,-2.5) {\rm row 2};
\node at (-1.5,-3.5) {\rm row 1};
\node at (-1.5,-4.5) {\rm row 0};

\cell315 \cell325 \cell336 \cell346 \cell356

\cell215 \cell225 \cell232  

\cell419 \cell429 \cell439\cell441 \cell453 \cell462 \cell472 \cell483 \cell493

\cell111 \cell121\cell131

\cell016\cell026\cell036

\cell51{$\infty$} \cell52{$\infty$}  \cell53{$\infty$} \cell54{$\infty$}  \cell55{$\infty$}  \cell56{$\infty$}  \cell57{$\infty$}  \cell58{$\infty$}  \cell59{$\infty$} 


\draw (0,-4)--(0,1)--(3,1)--(3,-2)--(5,-2)--(5,-3)--(9,-3);

\end{tikzpicture} 
\caption{A sorted tableau $\sigma$, with 
	$\perm_t(\sigma) = {3 \choose 2,1}_t {2 \choose 1,1}_t {4 \choose 2,2}_t$.}
\label{fig:perm}
\end{figure}


Our first main result is a ``compact'' formula for the modified Macdonald
polynomials  $\widetilde{H}_{\lambda}(X; q, t)$.  Before explaining
our result, we first
recall the combinatorial formula 
 of Haiman, Haglund, and Loehr \cite{HHL04}.
\begin{theorem}[\cite{HHL04}]\label{thm:HHL}
        The modified Macdonald polynomial
        $\widetilde{H}_{\lambda}(X;q,t)$
        is given by
\[
        \widetilde{H}_{\lambda}(X;q,t) = \sum_{\sigma\ :\ \dg(\lambda') \to \Z^+} x^\sigma q^{\inv(\sigma)} t^{\maj(\sigma)}.
\]
(Below we will be using the alternate form of this identity implied by the 
well-known fact that $\widetilde{H}_{\lambda}(X;q,t)=\widetilde{H}_{\lambda ^{\prime}}(X;t,q)$.)
\end{theorem}


While \cref{thm:HHL} is simple and elegant, it has the disadvantage of having many terms, since it is a sum over \emph{all} fillings of a diagram by positive integers. By contrast, our compact formula (\cref{thm:MacPerm2}) is a sum over far fewer terms -- it is a sum over \emph{sorted tableaux}.  To define these sorted tableaux,
we first define an order on the columns of the fillings.
\begin{definition}\label{def:orderhhl}
	Fix a filling $\sigma$ of $\dg(\lambda)$ and consider two columns $A$ and $B$ of height $j$ in $\lambda$.
Let $a_1,\dots, a_j$ and $b_1,\dots, b_j$ be the entries of columns $A$ and $B$, respectively, read from bottom to top.
We say that $A \lhd B$, if either $a_1 < b_1$, or 
	$a_i = b_i$ for $i=1, 2,  \dots, h-1$ (for some positive $h$), and 
	the cells containing $b_h$, $a_h$ and $a_{h-1}$ do not form an inversion triple.	
\end{definition}

\begin{definition}\label{def:ptableau2}
Given a filling $\sigma$ of $\dg(\lambda)$, we say that $\sigma$ is a \emph{sorted tableau} if, for all positive integers $h$, when we read all columns of height $h$ from left to right, the columns appear in weakly increasing order with respect to $\lhd$.  We write $\ST(\lambda)$ for the set of all sorted fillings of $\dg(\lambda)$.
\end{definition}


Let $\sigma$ be a sorted tableau. First suppose that the diagram of $\lambda$ is an $m \times n$ rectangle. The $n$ columns may not all have distinct fillings: suppose that among those $n$ columns, there are $j$ distinct column fillings, with $u_1$ identical columns of the first filling, $u_2$ identical columns of the second filling, \dots, $u_j$ identical columns all containing the $j$th filling. Define 
\[
\perm_t(\sigma) = {n \choose{u_1,\ldots,u_j}}_t.
\]
Suppose $\sigma$ is a sorted tableau which is a concatenation of rectangular sorted tableaux $\sigma_1,\dots, \sigma_{\ell}$, all of different heights.  Define $\perm_t(\sigma) = \prod_{i=1}^{\ell} \perm_t(\sigma_i).$ (See \cref{fig:perm}.)


Our main result in this section is a  compact formula for $\widetilde{H}_{\lambda'}$.
\begin{theorem}{\label{thm:MacPerm2}}
The modified Macdonald polynomial
$\widetilde{H}_{\lambda'}(X;q,t)$
	equals 
	\[
	\widetilde{H}_{\lambda}(X;q,t) = \sum_{\sigma\in \ST(\lambda)} x^\sigma t^{\inv(\sigma)} q^{\maj(\sigma)} 
	\perm_t(\sigma),
\]
where the sum is over all sorted fillings of $\dg(\lambda)$.
\end{theorem}






\begin{example} We use the theorem to compute
 $\widetilde{H}_{2,1,1}(x_1, x_2, x_3;q,t)$
 in \cref{fig:alternativeDef}.
\begin{figure}
\begin{center}
\resizebox{\linewidth}{!}{
\begin{tikzpicture}[node distance=2cm]
\def \sh {1.5};
\node at (0,0) {$\tableau{ 1\\1&1&1 }$};
\node at (0,-.75) {$1$};
\node at (\sh,0) {$\tableau{ 2\\1&1&1 }$};
\node at (\sh,-.75) {$q$};
\node at (2*\sh,0) {$\tableau{ 1\\2&1&1 }$};
\node at (2*\sh,-.75) {$t^2$};
\node at (3*\sh,0) {$\tableau{ 1\\1&1&2 }$};
\node at (3*\sh,-.75) {$[2]$};
\node at (4*\sh,0) {$\tableau{ 2\\2&1&1 }$};
\node at (4*\sh,-.75) {$t^2$};
\node at (5*\sh,0) {$\tableau{ 1\\1&2&2 }$};
\node at (5*\sh,-.75) {$1$};
\node at (6*\sh,0) {$\tableau{ 1\\2&1&2 }$};
\node at (6*\sh,-.75) {$t[2]$};
\node at (7*\sh,0) {$\tableau{ 2\\1&1&2 }$};
\node at (7*\sh,-.75) {$q[2]$};
\node at (8*\sh,0) {$\tableau{ 2\\2&1&2 }$};
\node at (8*\sh,-.75) {$t[2]$};
\node at (0,-2) {$\tableau{ 2\\1&2&2 }$};
\node at (0,-2.75) {$q$};
\node at (\sh,-2) {$\tableau{ 1\\2&2&2 }$};
\node at (\sh,-2.75) {$1$};
\node at (2*\sh,-2) {$\tableau{ 1\\1&2&3 }$};
\node at (2*\sh,-2.75) {$[2]$};
\node at (3*\sh,-2) {$\tableau{ 1\\2&1&3 }$};
\node at (3*\sh,-2.75) {$t[2]$};
\node at (4*\sh,-2) {$\tableau{ 2\\1&1&3 }$};
\node at (4*\sh,-2.75) {$q[2]$};
\node at (5*\sh,-2) {$\tableau{ 1\\3&1&2 }$};
\node at (5*\sh,-2.75) {$t^2[2]$};
\node at (6*\sh,-2) {$\tableau{ 3\\1&1&2 }$};
\node at (6*\sh,-2.75) {$q[2]$};
\node at (7*\sh,-2) {$\tableau{ 2\\3&1&1 }$};
\node at (7*\sh,-2.75) {$t^2$};
\node at (8*\sh,-2) {$\tableau{ 3\\2&1&1 }$};
\node at (8*\sh,-2.75) {$qt^2$};
\node at (0,-4) {$\tableau{ 2\\1&2&3 }$};
\node at (0,-4.75) {$q[2]$};
\node at (\sh,-4) {$\tableau{ 1\\2&2&3 }$};
\node at (\sh,-4.75) {$[2]$};
\node at (2*\sh,-4) {$\tableau{ 1\\3&2&2 }$};
\node at (2*\sh,-4.75) {$t^2$};
\node at (3*\sh,-4) {$\tableau{ 3\\1&2&2 }$};
\node at (3*\sh,-4.75) {$q$};
\node at (4*\sh,-4) {$\tableau{ 2\\2&1&3 }$};
\node at (4*\sh,-4.75) {$t[2]$};
\node at (5*\sh,-4) {$\tableau{ 1\\2&3&3 }$};
\node at (5*\sh,-4.75) {$1$};
\node at (6*\sh,-4) {$\tableau{ 2\\1&3&3 }$};
\node at (6*\sh,-4.75) {$q$};
\node at (7*\sh,-4) {$\tableau{ 3\\1&2&3 }$};
\node at (7*\sh,-4.75) {$q[2]$};
\node at (8*\sh,-4) {$\tableau{ 1\\3&2&3 }$};
\node at (8*\sh,-4.75) {$t[2]$};
\node at (9*\sh,-0) {$\tableau{ 2\\3&1&3 }$};
\node at (9*\sh,-0.75) {$t[2]$};
\node at (9*\sh,-2) {$\tableau{ 3\\2&1&3 }$};
\node at (9*\sh,-2.75) {$qt[2]$};
\node at (9*\sh,-4) {$\tableau{ 3\\3&1&2 }$};
\node at (9*\sh,-4.75) {$t^2[2]$};

\node at (10*\sh,-0) {$\tableau{ 2\\3&1&2 }$};
\node at (10*\sh,-0.75) {$t^2[2]$};
\node at (10*\sh,-2) {$\tableau{ 3\\2&1&2 }$};
\node at (10*\sh,-2.75) {$qt[2]$};
\end{tikzpicture}
}
\end{center}
        \caption{We compute $\widetilde{H}_{(2,1,1)}(x_1,x_2,x_3;q,t)$ by adding the weights of the sorted fillings of $\dg((2,1,1))$. In the figure above, we've listed $t^{\inv(\sigma)} q^{\maj(\sigma)} \perm_t(\sigma)$ below each tableau, but have omitted $x^{\sigma}$ to save space.  Here $[i]$ denotes $[i]_t$. }\label{fig:alternativeDef}
\end{figure}
\end{example}


To prove \cref{thm:MacPerm2}, we define \emph{inversion flip operators} $\mathcal{T}_i$ which act on fillings. These operators fix the $\maj$ statistic and change the $\inv$ statistic by one; in other words, they change the number of counterclockwise inversion triples  by one. Our operators are a generalization of the \emph{inversion flip} move introduced by Loehr and Niese~\cite{LoeNie12} to prove two-column recursions for Macdonald polynomials.  

Starting from a sorted tableau $\sigma$, we generate a family $\mathcal{F}(\sigma)$ of fillings by sequentially applying the operators $\mathcal{T}_i$. We then show that $$\sum_{\tau \in \mathcal{F}(\sigma)} x^{\tau}t^{\inv(\tau)}q^{\maj(\tau)} = x^{\sigma}t^{\inv(\sigma)}q^{\maj(\sigma)}\perm_t(\sigma)$$ and that $\displaystyle \bigsqcup_{\sigma\in\ST(\lambda)} \{\mathcal{F}(\sigma)\} =\bigcup_{\tau:\lambda \rightarrow \mathbb{Z}^+} \tau$, thus completing the proof.



\section{A compact formula for integral  Macdonald polynomials}\label{sec:Jcompact}

%\SM{In the formula below, is it ok to replace $Z$ (the two-variable alphabet) with $X$ (one variable)?}
In this section we provide a compact formula for the integral form 
Macdonald polynomials $J_{\mu}(X;q;t)$.
We use notational conventions from \cite[Appendix A]{Hag08}; in particular, the reader should consult that source for the
definition of type A and type B inversion and coinversion triples.  Recall that for any composition $\alpha$, $\dg(\alpha)$ refers to the 
diagram with $\alpha _i$ squares in column $i$.  We first recall the formula for $J_{\mu}(X;q,t)$ from \cite{HHL04}. 

\begin{theorem}[\cite{HHL04}]\label{thm:4.1}
	The integral form Macdonald polynomial is given by 
\begin{align} 
\label{Jmu3}
J_{\mu}(X;q,t) =
(1-t)^{\ell (\mu)} \sum_{ \text{nonattacking fillings $\sigma$ 
of $\dg(\mu )$ }} x^{\sigma}
q^{\maj(\sigma)} 
t^{\coinv(\sigma)} \\
\notag
\times \prod _{ \substack{s \in \dg(\mu)\\ \text{$s$ not in row $1$}\\ \sigma (s) = \sigma (\South(s))} }
(1- q^{\text{leg}(s)+1}t^{\arm(s)+1})
\prod _{ \substack{s \in \dg(\mu ) \\ \text{$s$ not in row $1$}\\ \sigma (s) \ne \sigma ( \South(s) ) } }(1-t),
\end{align}
where $\ell (\mu)$ is the number of parts of $\mu$, and for a square $s$ not in row $1$,  
$\South(s)$ denotes the square directly below 
$s$ in the same column as $s$.  Here the sum is over all nonattacking filings of $\dg(\mu )$ (there is no basement in these fillings).
\end{theorem}

In \cite{HHL04} the authors note that the right-hand-side of \eqref{Jmu3} actually yields a correct formula for $J_{\mu}$ if we replace $\mu$ everywhere by $\alpha$, where $\alpha$ is any weak composition of $n$ into $n$ parts satisfying $\dec(\alpha) = \mu$.  (See~\cite[Appendix A]{Hag08} for details about how to extend the notion of $\coinv$ to increasing column heights using type $B$ triples.)  In fact, the most efficient formula for $J_{\mu}$ seems to be the case where $\alpha = \inc(\mu)$.

One unpleasant feature of all these formulas for $J_{\mu}$  is that for the special case $\mu=1^n$, 
\begin{align}
\label{simple}
J_{1^n}(X;q,t)=x_1x_2\cdots x_n (1-t)(1-t^2)\cdots (1-t^n),
\end{align}

while the formula from \eqref{Jmu3} reduces to
\begin{align}
 x_1x_2\cdots x_n (1-t)^n \sum_{\sigma \in S_n}t^{\coinv(\sigma)},
\end{align}
a sum of $n!$ terms.  In this section we show how the identity
\begin{align}
\label{newPmu}
P_{\mu}(X;q,t) =
\sum_{\alpha:\ \dec(\alpha)= \mu} E_{ \inc(\alpha) }^{ \beta (\alpha) }( X;q,t),
\end{align}
yields a corresponding formula for $J_{\mu}$ which, when applied to the case $\mu=1^n$, has only one term - identity \eqref{simple}.  We mention that \eqref{newPmu}
is closely related to Lenart's formula for $P_{\mu}$ \cite{Lenart} 
(which he proved under the additional assumption that $\mu$ has distinct parts).

\begin{definition}   Let $\alpha$ be a weak composition of $n$ into $n$ parts such that $\inc(\alpha) = \alpha$.   We say a nonattacking filling $\sigma$ of $ \dg(\alpha )$ 
(with or without a basement) is \emph{ordered} if in the bottom row of  $\dg(\alpha )$, entries of $\sigma$ below
columns of the same height are strictly decreasing when read left to right.  We call $\OF(\alpha)$ the set of ordered, nonattacking fillings of $\dg(\alpha)$. 
\end{definition}
	\cref{nonatt3b} shows an ordered nonattacking filling.  The $7$ coinversion triples for the filling in \cref{nonatt3b} are 
$(1,6,7), (3,6,7), (5,6,7), (6,7,9), (1,2,3),(1,2,9), (3,5,7)$.

\begin{figure}
\begin{center}
\begin{tikzpicture}[scale=.5]
\cell043
\cell147 \cell132 \cell121 \cell 115
\cell246 \cell231 \cell223 \cell215 \cell204
\end{tikzpicture}

\caption{An ordered nonattacking filling of $\dg((0,0,1,2,2,2,3))$ with $\maj=3$ and $\coinv= 7$.}
\label{nonatt3b}
\end{center}
\end{figure}

Recall that Macdonald's definition of the integral form $J_{\mu}(X;q,t)$ is
\begin{align}
\label{JDef}
J_{\mu}(X;q,t) &= P_{\mu}(X;q,t) \text{PR}1(\mu),
\end{align}
where 
\begin{align}
\label{PR1}
\text{PR}1(\mu) = \prod _{s \in \dg(\mu')} (1-q^{\arm(s)}t^{\text{leg}(s)+1}) = \prod _{s \in \dg(\mu )} (1-q^{\text{leg}(s)}t^{\arm(s)+1}).
\end{align}

For a composition $\alpha$, define $m_i(\alpha)$ to be the number of times $i$ occurs in $\alpha$ for $i\ge 1$. 

\begin{proposition} 
\label{13}
For $\alpha$ a weak composition,  define
\begin{align}
\label{EQ}
\text{PR}2(\alpha) =  \prod_{i\ge 1} (t;t)_{m_i}
\prod_{\substack{s \in \dg(\inc(\alpha) )\\ \text{$s$ not in the bottom row} }  } (1-q^{\text{leg}(s)+1} t^{\arm(s)+1}),
\end{align}
where $m_i=m_i(\alpha)$, and the statistics $\arm$ and $\leg$ are defined as in \cref{sec:def}. Then if $\mu$ is any partition, $\text{PR}1(\mu) = \text{PR}2(\inc(\mu))$.
\end{proposition}

\begin{definition}
Given a composition $\alpha$ of $n$ into $n$ parts, we define the {\it integral form} version of $E_{\inc(\alpha)} ^{\beta (\alpha)}(X;q,t)$ as
\begin{align}
\label{integralE}
\mathcal {E}_{\inc(\alpha)} ^{\beta (\alpha)}(X;q,t) = PR2(\inc(\alpha))E_{\inc(\alpha)} ^{\beta (\alpha)}(X;q,t).
\end{align}
\end{definition}

Recall the following combinatorial formula 
for $E_{\alpha}^{\sigma}(X;q,t)$ in
\cite{Al16};
\begin{align}
\label{formula}
E_{\inc(\alpha)}^{\beta (\alpha)}(X;q,t) = \sum_{\sigma} x^{\sigma} \text{wt}(\sigma),
\end{align}
where the sum is over all nonattacking fillings $\sigma$ of $\dg(\alpha)$ with basement $\beta (\alpha)$.  The weight of such a filling is
\begin{align}
\label{weight}
\text{wt}(\sigma) = q^{ \maj (\sigma)} t^{ \coinv(\sigma) }\prod_{s:\ \sigma(s) \ne \sigma(\South(s))} \frac {1-t}{1-q^{\text{leg(s)}+1}t^{\text{arm(s)}+1}}.
\end{align}

For the following, we set $m_i=m_i(\alpha)$. It follows from the above formula that $\mathcal {E}_{\inc(\alpha)} ^{\beta (\alpha)}(X;q,t)$ 
is $\prod_{i} (t;t)_{m_i}$ times an element of $\mathbb Z [x_1,\ldots ,x_n,q,t]$.  To see this, note that every nonattacking filling $\sigma$ of $\inc(\alpha)$ has the property that each entry in the bottom row is equal to the entry in the basement directly below it, and hence doesn't contribute anything to the product in \eqref{weight} defining $\text{wt}(\sigma)$, while if any entry above the bottom row satisfies $\sigma (s) \not= \sigma (\South(s))$, then the associated  factor  
$(1-q^{\text{leg}(s)+1}t^{\arm(s)+1})$ is the exact term in the coefficient of
 $E_{\inc(\alpha)} ^{\beta (\alpha)}(X;q,t)$ from \eqref{integralE} above corresponding to square $s$.    In fact this argument shows that
\begin{align}
\label{IFE}
\mathcal {E}_{\inc(\alpha)} ^{\beta (\alpha)}(X;q,t) = 
\prod_{i}(t;t)_{m_i}  
\sum_{\substack{\sigma\in\OF(\inc(\alpha))\\
\text{with basement $\beta (\alpha)$} } }
x^{\sigma}q^{\maj(\sigma) } t^{\coinv(\sigma )} \\
\notag
\times \prod _{\substack{s \in \dg(\inc(\alpha) ) ,\ \text{$s$ not in row $1$} \\
\sigma (s) = \sigma (\South(s))} }
(1- q^{\text{leg}(s)+1}t^{\arm(s)+1})
\prod _{ \substack{s \in \dg(\inc(\alpha) ),\ \text{$s$ not in row $1$}\\ \sigma (s) \ne \sigma ( \South(s) ) } }(1-t),
\end{align}

\begin{corollary} The formula for $J_{\mu}$ has the following more compact version.
\begin{align} 
\label{Jmu4}
J_{\mu}(X;q,t) =
\prod_{i}(t;t)_{m_i}  
\sum_{ \sigma\in \OF(\mu) } 
x^{\sigma}q^{\maj(\sigma)} t^{\coinv(\sigma)} \\
\notag
\times \prod _{\substack{s \in \dg(\mu ),\ \text{$s$ not in row $1$}\\ \sigma (s) = \sigma (\South(s))} }
(1- q^{\text{leg}(s)+1}t^{\arm(s)+1})
\prod _{ \substack{s \in \dg(\mu ),\ \text{$s$ not in row $1$}\\ \sigma (s) \ne \sigma ( \South(s) ) } }(1-t),
\end{align}
where $m_i=m_i(\mu)$.
\end{corollary}

\begin{proof}
Start by multiplying both sides of  \eqref{newPmu} by $\text{PR}2(\inc(\mu))$.  The left-hand-side then becomes $J_{\mu}(X;q,t)$ by \eqref{JDef} and \cref{13}.   
The summand on the right hand side becomes $\mathcal {E}_{\inc(\alpha)} ^{\beta (\alpha)}(X;q,t)$, which by \eqref{IFE} equals the portion of \eqref{Jmu4}
which has bottom row determined by $\beta(\alpha)$.
\end{proof}

Note that \eqref{Jmu4} implies the (as far as we know) new fact that $J_{\mu}(X;q,t)$ is $\prod_{i}(t;t)_{m_i}$ times an element of $\mathbb Z [x_1,\ldots, x_n,q,t]$.


\section{A quasisymmetric Macdonald polynomial}\label{sec:qsym}

Recall that the ring of \emph{quasisymmetric functions} is a graded ring 
which  contains within it 
the ring of symmetric functions.  
The ring of quasisymmetric functions
 has multiple
distinguished bases, indexed by (strong) compositions.  
One such basis is the \emph{quasisymmetric Schur
functions} 
$\text{QS}_{\gamma}(X)$ 
introduced by the second and fourth authors, together with Luoto and van Willigenburg 
\cite{HLMV09}.  
The authors showed that $\text{QS}_{\gamma}(X)$ is quasisymmetric,
and that each (symmetric) Schur function $s_{\lambda}(X)$ 
is a positive sum of quasisymmetric 
Schur functions.  In light of this, and the fact that Macdonald 
polynomials expand positively in terms of Schur polynomials, it is 
natural to ask if there is a notion of a 
\emph{quasisymmetric Macdonald polynomial} $G_{\gamma}(X; q, t)$ such that:
\begin{enumerate}
\item[(A)]\label{a} The symmetric Macdonald polynomial $P_{\lambda}(X; q, t)$ is a positive sum of quasisymmetric Macdonald polynomials;     \item[(B)]\label{b} $G_{\gamma}(X; q, t)$ is quasisymmetric; 
 \item[(C)]\label{c} $G_{\gamma}(X; 0, 0)$ is the quasisymmetric Schur function 
		$\text{QS}_{\gamma}(X)$.
 \item[(D)] \label{d} $G_{\gamma}(X; q, t)$ has a combinatorial formula
along the lines of the ``HHL'' formula for the $E_{\alpha}$ \cite{HHL08}, or its compact version from \cite{CMW18}.
\end{enumerate}
We show in this section that the answer to this question is yes.


Given a permutation $\tau \in S_n$,  let $E_{\alpha}^{\tau}(X;q,t)$ be the
 permuted-basement nonsymmetric Macdonald polynomial defined in \cite{thesis} and studied in  \cite{Al16,CMW18}, and let
 $F_{\alpha}(X;q,t) = E_{\inc(\alpha)}^{\beta (\alpha)}(X;q,t)$.  For any partition $\lambda$ of $n$, from \cite{CMW18} we have that 
 \begin{align}
 \label{basic}
 P_{\lambda}(X;q,t)  = \sum_{\alpha :\ \dec(\alpha)= \lambda} F_{\alpha}(X;q,t),
 \end{align}
where the sum is over all weak compositions $\alpha$ whose positive parts are a 
rearrangement of the parts of $\lambda$. 

Note that if $\text{id}=(1,2,\ldots ,n)$ and  $w_0=(n,n-1,\ldots,1)$ are the identity permutation and permutation of maximal length in $S_n$,
respectively, then
$E_{\alpha}^{ \text{id} }(X;0,0)$ is the Demazure atom and $E_{\alpha}^{ w_0 }(X;0,0)$ the Demazure character. (In the common notation for Demazure characters, 
i.e. key polynomials, one reverses the vector $\alpha$, i.e.  the key polynomial corresponding to $\alpha$ would be $E_{(\alpha _n, \ldots ,\alpha _1)}^{w_0}(X;0,0)$.)

Motivated by \eqref{basic}, we have the following definition and theorem.

\begin{theorem}
	We define the \emph{quasisymmetric Macdonald polynomial}
	$G_{\gamma}(X;q,t)$  to be
\begin{align}
\label{Gdef}
G_{\gamma}(X;q,t) &= \sum_{\alpha:\ \alpha ^{+}= \gamma}  F_{\alpha}(X;q,t) \\
&=  \sum_{\alpha:\ \alpha ^{+}= \gamma}  E_{\inc(\alpha)}^{\beta (\alpha )}(X;q,t),
\end{align}
where the sum is over all weak compositions $\alpha$ for which $\alpha^{+}=\gamma$.
	Then $G_{\gamma}(X; q, t)$ satisfies properties 
	\rm{(A), (B), (C), and (D)}.
\end{theorem}

The fact that $G_{\gamma}(X; q, t)$ satisfies \rm{(A)} follows from 
 \eqref{basic}.
There are several combinatorial proofs that $G_{\gamma}(X; q,t)$ is quasisymmetric and hence satisfies
\rm{(B)}.  
One proof uses
\cref{formula} and a notion of \emph{packed} nonattacking fillings.
Another proof uses the multiline queues from \cite{CMW18}.
To see that  $G_{\gamma}(X; q, t)$ satisfies \rm{(C)}, recall that for $\gamma$ a strong composition of $n$, 
$\text{QS}_{\gamma}$ is defined by the equation
\begin{align}
\label{QS}
\text{QS}_{\gamma}(X) = \sum_{\alpha:\ \alpha ^{+}= \gamma} E_{\alpha}^{\text{id}}(X;0,0).
\end{align}
So to verify \rm{(C)}, it suffices to show that 
$F_{\alpha}(X; 0, 0) = E_{\alpha}^{\text{id}}(X; 0, 0)$.  We actually show the stronger statement
that 
\begin{equation} \label{tatom}
	F_{\alpha}(X; 0, t) = E_{\alpha}^{\text{id}}(X; 0, t),
\end{equation}
	where $E_{\alpha}^{\text{id}}(X; 0, t)$ is the Demazure $t$-atom.
To prove this, one can use induction, together with the action of the Hecke operators.
In particular, by \cite{CMW}, $T_i F_{\alpha} = F_{s_i \alpha}$ if $\alpha_i >\alpha_{i+1}$; compare with 
\cite[Corollary 26]{Al16}.

\begin{comment}
From \cite{HHL08} we have
\begin{align}\label{PtoE}
P_{\lambda}(X;q,t)  = \sum_{\alpha: \alpha ^{+} =\lambda} c_{\lambda,\alpha} E_{\alpha}^{id}(X;q,t),
\end{align}
where $c_{\lambda,\alpha}$ is a certain explicit rational function in $q, t$.  Letting $q=t=0$ in this identity, since $c_{\lambda}(0,0)=1$, we get the formula of Lascoux-Sch\"utzenberger expressing the Schur function $s_{\lambda}$ as a sum of Demazure atoms. In view of~\eqref{QS}, this also shows that the Schur function is a sum of the $\text{QS}_{\gamma}$.  Trying the same idea in \eqref{PtoE} though, one finds the sum of the $c_{\lambda,\alpha}E_{\alpha}^{id}(X;q,t)$ over $\alpha^{+}=\gamma$ are not generally quasisymmetric.  

We first note that $G_{\gamma}$ has the desired property of specializing to $\text{QS}_{\gamma}$ at $q=t=0$.
\begin{proposition}
For a composition $\gamma$, we have
\[
G_{\gamma}(X;0,0)=QS_{\gamma}(X).
\]
\end{proposition}

We now show that 
\begin{theorem}
$G_{\gamma}(X;q,t)$ is a quasisymmetric function.
\end {theorem} 

\begin{remark}
\label{non}
For general diagrams, the product in \eqref{weight} normally will include squares $s$ in row $1$, 
but since the parts of  $\inc(\alpha)$ are by definition in nondecreasing order, the nonattacking condition forces the entries in row $1$ to be the same as those
directly below them in the basement.  Hence for diagrams corresponding to partitions, we can restrict the product in \eqref{weight} to squares $s$ above row $1$.    
Furthermore, since $\sigma (s)$ for squares $s$ in row $1$ equals $\sigma (\South(s))$, these pairs of equal squares will not form any coinversion triples with 
any third squares in the arm of $s$, and will also not contribute to $\maj$.
\end{remark}


\begin{figure}
\begin{center}
\begin{tikzpicture}[scale=.5]
\cell041 \cell141 \cell136 \cell122
\cell243 \cell234 \cell223
\cell343 \cell338 \cell323
 \cell444 \cell435 \cell424 \cell419
\cell5{-8}{13} \cell5{-7}{12} \cell5{-6}{11} \cell5{-5}{10} \cell5{-4}7 \cell5{-3}6 \cell5{-2}3 \cell5{-1}2 \cell501 \cell519 \cell528 \cell535 \cell544
\end{tikzpicture}\hspace{.5cm}
\begin{tikzpicture}[scale=.5]
\cell041 
\cell141 \cell139 \cell123
\cell245 \cell236 \cell22{10}
\cell345 \cell33{11} \cell323
\cell446 \cell438 \cell42{11} \cell41{13}
\cell5{-8}{12} \cell5{-7}{10} \cell5{-6}{9} \cell5{-5}7 \cell5{-4}5 \cell5{-3}4 \cell5{-2}3 \cell5{-1}2 \cell501 \cell51{13} \cell52{11} \cell538 \cell546
\end{tikzpicture}
\caption{On the left, a packed, nonattacking filling.  
On the right, a nonattacking filling with the same weight.}\label{nonatt1b}
\end{center}
\end{figure}

Call a nonattacking filling $\sigma$ {\it packed} if the set of numbers occurring above the basement in $\sigma$ equal $\{1,2,\ldots ,k\}$ for some $k$, as in
\cref{nonatt1b} (where $k=9$).
By \cref{non} any entry in the bottom row will not generate any contribution to $\text{wt}(\sigma)$.   The contribution to 
$\text{wt}(\sigma)$ by the numbers in the rows above depends only on the
relative order of these numbers; hence  if we replace the filling $\sigma$ by another filling $R$ using a replacement of the form $i \to w_i$ for $1\le i \le k$, where
$k$ is the maximal value occurring above the basement in $\sigma$  and $1\le w_1 < w_2 < \cdots < w_k\le n$, then the vector of exponents $V(x^{R})$ satisfies
$V(x^{R})^{+} = V( x^{\sigma})^{+}$, and $\text{wt}(\sigma) = \text{wt}(R)$.
For example, the monomial weight of the packed filling on the left of \cref{nonatt1b} is $x_1^2x_2^2x_3^2x_4^2x_5x_6x_7x_8^2x_9$, 
with vector $V = (2,2,2,2,1,1,1,2,1,0,0,0,0)$, while the monomial weight of the filling on the right of \cref{nonatt1b}
is $x_1^2 x_3^2 x_5^2 x_6^2 x_8 x_9 x_{10} x_{11}^2 x_{13}$, with vector $V=(2,0,2,0,2,2,0,1,1,1,2,0,1,0)$. 
Hence each packed nonattacking filling can be associated to a family of nonattacking fillings with the 
same $\text{wt}$.  The sum of the monomial weights of all these fillings is a monomial quasisymmetric function by definition, and it follows that $G_{\gamma}(X;q,t)$ is
quasisymmetric.
\end{comment}


It would be interesting to find a connection between the quasisymmetric Macdonald polynomials
$G_{\gamma}(X; q, t)$ that we introduce in this paper, and other objects in the literature.
We note that 
our $G_{\gamma}$ are different from the duals of the noncommutative symmetric function analogues of Macdonald polynomials introduced in~\cite{BerZab05}; 
we also do not see a connection to the noncommutative Hall-Littlewood polynomials studied in~\cite{Hiv00}.

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