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%% Theorem definitions
\theoremstyle{plain}
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\newtheorem{theorem*}{Theorem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{lemma}[theorem]{Lemma}
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\crefname{lemma}{Lemma}{Lemmas}

\theoremstyle{definition}
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%%% tableau macros %%%

\newlength\cellsize \setlength\cellsize{12\unitlength}
\savebox2{%
\begin{picture}(12,12)
\put(0,0){\line(1,0){12}}
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\end{picture}}
\newcommand\cellify[1]{\def\thearg{#1}\def\nothing{}%
\ifx\thearg\nothing
\vrule width0pt height\cellsize depth0pt\else
\hbox to 0pt{\usebox2\hss}\fi%
\vbox to 12\unitlength{
\vss
\hbox to 12\unitlength{\hss$#1$\hss}
\vss}}
\newcommand\tableau[1]{\vtop{\let\\=\cr
\setlength\baselineskip{-16000pt}
\setlength\lineskiplimit{16000pt}
\setlength\lineskip{0pt}
\halign*{&\cellify{##}\cr#1\crcr}}}
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\begin{picture}(12,12)
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\end{picture}}
\newcommand\expath[1]{%
\hbox to 0pt{\usebox3\hss}%
\vbox to 12\unitlength{
\vss
\hbox to 12\unitlength{\hss$#1$\hss}
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\newcommand\cell[3]{
\def\i{#1} \def\j{#2} \def\entry{#3}
\draw (\j-1,-\i)--(\j,-\i)--(\j,-\i+1)--(\j-1,-\i+1)--(\j-1,-\i);
\node at (\j-.5,-\i+.5) {\entry};
}

\newcommand\circleT[5]{
\def \n {5}
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\def \margin {20}
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      \xdef\s{\pgfmathresult}
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}

%%%%% end tableaux macros %%%%%%%%%%%%%%%%%%%%%%%%%%%%


\renewcommand{\arraystretch}{2}

%\usepackage{lipsum}

%% define your title in the usual way
\title[Quasisymmetric Macdonald]{Fundamental expansion of 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, O. Mandelshtam, and A. Roberts]{
Sylvie Corteel\thanks{\href{mailto:corteel@berkeley.edu}{corteel@berkeley.edu} }%Sylvie Corteel} 
\addressmark{1}, 
Olya Mandelshtam\thanks{\href{mailto:olya@math.brown.edu}{olya@math.brown.edu} OM was supported by NSF grants DMS-1704874 and DMS-1953891}\addressmark{2},
\and
Austin Roberts\thanks{\href{mailto:aroberts@highline.edu}{aroberts@highline.edu}}% Austin Roberts} 
\addressmark{3}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics, UC Berkeley, USA \\
\addressmark{2}Department of Mathematics, Brown University, USA \\
\addressmark{3}Department of Mathematics, Highline College, USA}

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

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

%% put your English abstract here, or comment this out if you don't have one yet
%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{The quasisymmetic Macdonald polynomials $G_{\gamma}(X;q,t)$ were recently introduced by the first and second authors with Haglund, Mason, and Williams to refine the symmetric Macdonald polynomials $P_{\lambda}(X;q,t)$. % with the property that $G_{\gamma}(X;0,0)$ equals $QS_{\gamma}(X)$, the quasisymmetric Schur polynomial of \cite{HLMvW11}. 
We derive an expansion for $G_{\gamma}(X;q,t)$ in the fundamental basis of quasisymmetric functions.}

%% put your French abstract here, or comment this out if you don't have one
%\resume{Frenchify this!!!! The quasisymmetic Macdonald polynomials $G_{\gamma}(X;q,t)$ were recently introduced by the first and second authors with Haglund, Mason, and Williams to refine the symmetric Macdonald polynomials $P_{\lambda}(X;q,t)$. % with the property that $G_{\gamma}(X;0,0)$ equals $QS_{\gamma}(X)$, the quasisymmetric Schur polynomial of \cite{HLMvW11}. 
%We derive an expansion for $G_{\gamma}(X;q,t)$ in the fundamental basis of quasisymmetric functions.}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{quasisymmetric, Macdonald polynomials, fundamental basis}

%% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easier
%% regardless of how you do it, please use numerical citations, ie. [xx, yy] in the text

%% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex
%% but you can use bibtex if you want
\usepackage[backend=bibtex]{biblatex}
\addbibresource{09Roberts.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 symmetric \emph{Macdonald polynomials} $P_{\lambda}(X; q, t)$ \cite{Macdonald} are a family of functions in $X = \{x_1, x_2,\dots \}$ indexed by partitions, whose coefficients depend on two parameters $q$ and $t$. %They can be defined as the unique monic basis for the ring of symmetric functions that satisfies certain triangularity and orthogonality conditions. Macdonald polynomials generalize multiple important families of polynomials, including Schur polynomials and Hall--Littlewood polynomials.
% 
The related \emph{nonsymmetric Macdonald polynomials} $E_{\mu}(X;q,t)$ were introduced shortly after as a tool to study Macdonald polynomials, in a series of papers by Cherednik \cite{Cher1}, Macdonald \cite{Mac95}, and Opdam \cite{Opd95}.
% first appeared shortly after the introduction of Macdonald polynomials 
The polynomials $E_{\mu}(X;q, t)$ are indexed by weak compositions and form a basis for the full polynomial ring $\mathbb{Q}[X](q,t)$. Ferreira \cite{Fer11} and later Alexandersson \cite{Ale16} studied the extension of these to the more general \emph{permuted basement} nonsymmetric Macdonald polynomials $E^\sigma_{\mu}(X;q,t)$, where $X= \{x_1, \ldots, x_n\}$,  $\sigma\in S_n$, and the length of $\mu$ is $n$. 


The combinatorics of Macdonald polynomials has been actively studied for decades. In \cite{HHL05}, Haglund, Haiman, and Loehr gave a combinatorial formula for the \emph{modified Macdonald polynomials},  $\widetilde{H}_{\lambda}(X;q,t)$, and %. A formula for 
the \emph{integral form},  $J_{\lambda}(X; q,t)$. %, was then given in \cite{HHL08}. 
%Important to our purposes, this paper also
In \cite{HHL08} they subsequently provided a formula for the nonsymmetric Macdonald polynomials $E_{\mu}(X;q,t)$, which was then broadened to the more general polynomials  $E^\sigma_{\mu}(X;q,t)$ in \cite{Ale16,Fer11}. 


In \cite{CHMMW}, the first and second authors with Haglund, Mason, and Williams introduced a new family of quasisymmetric functions $G_{\gamma}(X; q, t)$ they named \emph{quasisymmetric Macdonald polynomials}.  They showed that $G_{\gamma}(X; q, t)$ is indeed a quasisymmetric function, and gave a combinatorial formula for $G_{\gamma}(X; q, t)$ refining the compact formula for $P_{\lambda}$ from \cite{CMW18}. The Macdonald polynomial $P_{\lambda}(X; q, t)$ is a sum of these quasisymmetric Macdonald polynomials, and at $q=t=0$, $G_{\gamma}(X; q, t)$ specializes to the \emph{quasisymmetric Schur functions} $\text{QS}_{\gamma}(X)$ introduced by Haglund, Luoto, Mason, and van Willigenburg in \cite{HLMvW11}.  

The goal of this article is to write an expansion of the polynomials $G_{\gamma}(X; q, t)$ in the fundamental basis. This basis was introduced by Gessel in \cite{Ges84} and is one of the most common bases of the vector space of quasisymmetric functions. Our main results are the following Theorems, see \cref{sec:def} for the relevant definitions.
\begin{theorem}\label{thm:q}
Let $\gamma$ be a strong composition. Then
\begin{align*}
G_{\gamma}(X;q,t)= & \sum_{\tau\in\ST(\gamma)} t^{\coinv(\tau)}q^{\maj(\tau)} \left( \prod_{\substack{u\in\widehat{\dg}(\gamma)\\u\not\in W(\tau)}} \frac{1-t}{1-q^{\leg(u)+1}t^{\arm(u)+1}} \right) \\
&\times \sum_{U \subseteq W(\tau)} (-t)^{|U|} \left( \prod_{u \in U} \frac{1-q^{\leg(u)+1}t^{\arm(u)}}{1-q^{\leg(u)+1}t^{\arm(u)+1}} \right)F_{V(\tau)\cup U}.
\end{align*}\label{main}
\end{theorem}

\begin{theorem}\label{thm:HL}
%Let $\gamma$ be a strong composition.
%\[
%G_{\gamma}(X;0,t) = \sum_{\tau\in\ST_0(\gamma)} t^{\coinv(\tau)}(1-t)^{h(\gamma)-|W(\tau)|} \sum_{U\subseteq W(\tau)} (-t)^{|U|} F_{V(\tau)\cup U}
%\]
Let $\gamma$ be a strong composition. Then
\begin{align*}
G_{\gamma}(X;0,t)& =
\sum_{\tau\in\ST_{1}(\gamma)}
(1-t)^{\omega(\tau)}
(-t)^{|\Des(\tau)|}
t^{\coinv(\tau)-\coinv(\Des(\tau))} 
 F_{\Nu(\tau)}.
\end{align*}

\end{theorem}


This article proceeds through a series of purely combinatorial proofs and results using a variety of tableaux enumeration techniques, organized as follows.
%This paper is organized as follows. 
In \cref{sec:def}, we provide the relevant background. \cref{sec:packed} provides a proof for \cref{thm:q}. In \cref{sec:HL} we provide an alternative expansion in the Hall--Littlewood case, yielding \cref{thm:HL} and a related result for Jack polynomials.


\section{Preliminaries and definitions}
\label{sec:def}


For a nonnegative integer $n$, a \emph{weak composition} $\alpha=(\alpha_1,\ldots,\alpha_k) \models n$ is a list of nonnegative integers called the \emph{parts} of $\alpha$, summing to $n$, so that $n=|\alpha|=\sum_{i=1}^k \alpha_i$.  Let $\alpha^+$ denote the composition obtained by collapsing the (weak) composition $\alpha$ by removing the zero-parts from $\alpha$. We call a composition with no non-zero parts a \emph{strong composition}. If $\alpha_1\geq \alpha_2 \geq \cdots \geq \alpha_k$, then $\alpha$ is called a \emph{partition}. We denote by %$\dec(\alpha)$ the partition obtained by sorting the parts of $\alpha$ in decreasing order, and 
$\inc(\alpha)$ the composition obtained by sorting the parts of $\alpha$ in increasing order. Define $\beta(\alpha)$ to be the permutation of \emph{longest length} such that $\beta(\alpha) \circ \alpha = \inc(\alpha)$, where the length of a permutation is the number of inversions in its word representation.

\begin{example}
For $\alpha=(2,1,0,0,3,0,1)$, we have $\alpha^+=(2,1,3,1)$, %$\dec(\alpha)=(3,2,1,1,0,0,0)$, 
$\inc(\alpha)=(0,0,0,1,1,2,3)$, and $\beta(\alpha)=(6,4,3,7,2,1,5)$.
\end{example}

%%For two compositions, $\alpha,\beta$, we say $\beta$ is a \emph{refinement} of $\alpha$, denoted by $\beta<\alpha$, if $\alpha$ can be obtained by adding together adjacent parts of $\beta$. For example, $(2,1,3,1)<(2,4,1)<(2,5)<(7)$. 
%Finally, there is a natural bijection from (strong) compositions $\alpha\models n$ with $|S|+1$ parts to subsets $S\in [n-1]$, given by taking the difference between successive elements of $\{0,n\}\cup{S}$, after elements of this set are listed in order. \OM{is this needed?}%%Specifically, the subset $S$ corresponding to a composition $\alpha=(\alpha_1,\ldots,\alpha_k)$ is
%%\[
%%S = \{\alpha_1,\alpha_1+\alpha_2,\ldots,\alpha_1+\alpha_2+\cdots+\alpha_{k-1}\},
%%\]
%%and the composition $\alpha\models n$ corresponding to a subset $S=\{i_1,i_2,\ldots,i_{k-1}\}\subseteq [n-1]$ is
%%\[
%%\alpha=(i_1,i_2-i_1,i_3-i_2,\ldots,n-i_{k-1}).
%%\]

%\begin{example}
%$\alpha=(2,1,3,2)$ corresponds to the subset $S=\{2,3,6\}\subseteq [7] $. 
%\end{example}

\subsection{Quasisymmetric functions}
%The vector space of \emph{quasisymmetric functions} properly contains the space of symmetric functions. A quasisymmetric function is a bounded degree formal power series $f\in \mathbb{F}[x_1,x_2,\ldots]$ such that for all $k$, all compositions $\alpha=(\alpha_1,\alpha_2,\ldots,\alpha_k)$, and all sets of indices $i_1<i_2<\cdots<i_k$, the coefficients of $x_1^{\alpha_1}\ldots x_k^{\alpha_k}$ and $x_{i_1}^{\alpha_1}\ldots x_{i_k}^{\alpha_k}$ in $f$ are equal. 
%
Similar to the symmetric functions, the vector space of \emph{quasisymmetric functions} has several natural bases consisting of functions of fixed degree. We will focus on the \emph{monomial basis} $\{M_S\}$ and the \emph{fundamental basis} $\{F_S\}$, indexed by subsets $S \subset [n-1]$, for each fixed degree $n$. The monomial basis functions are defined as
%For each fixed degree $n$, the elements of the \emph{monomial basis} are indexed by subsets $S \subset [n-1]$. 
\begin{equation}
M_S := \sum_{i_1 < i_2 < \cdots < i_{k}} x_{i_1}^{\alpha_1}x_{i_2}^{\alpha_2}\cdots x_{i_k}^{\alpha_k}
\end{equation}
where $k=|S|+1$, and $\alpha$ is the (strong) composition corresponding to the subset $S$.

The fundamental basis functions are defined as
%Another important basis is the \emph{fundamental quasisymmetric functions}, similarly indexed by $S\subset [n-1]$,
\begin{equation}
F_S := \sum_{i_1 \leq i_2 \leq \cdots \leq i_n \atop  j \in S \implies i_j \neq i_{j+1}} x_{i_1}x_{i_2}\cdots x_{i_n}.
\end{equation}

\noindent
For example, 
%\begin{align*}
\[M_{\{2,3,6\}} %&
= \sum_{i_1 < i_2 < i_3 < i_{4}} x_{i_1}^{2}x_{i_2}^{1}x_{i_3}^3 x_{i_4}^{2},%\\
\quad\mbox{and}\quad F_{\{2,3,6\}}%&
=\sum_{i_1 \leq i_2 < i_3 < i_4\leq i_5\leq i_6<i_7\leq i_8} x_{i_1}x_{i_2}\cdots x_{i_8}.
%\end{align*}
\]

Let $S\subseteq [n-1]$. It follows that 
\begin{equation}\label{eq:F}
F_{S}=\sum_{S\subseteq S'} M_{S'}.
\end{equation}


\noindent
For example, let $n=8$ and $S=\{1,4\}$. Then 
\begin{align*}
F_{\{1,4\}} &= M_{\{1,4\}}+M_{\{1,2,4\}}+M_{\{1,3,4\}}+M_{\{1,2,3,4\}}.
\end{align*}





The goal of this article is to give an expansion of the quasisymmetric Macdonald polynomial $G_{\gamma}(X;q,t)$, which we present below, in terms of the fundamental quasisymmetric basis. Let $\gamma$ be a strong composition. The quasisymmetric Macdonald polynomial is defined by the infinite sum
\begin{align}\label{def: Mac}
G_{\gamma}(X;q,t) =  \sum_{\alpha:\ \alpha ^{+}= \gamma}  E_{\inc(\alpha)}^{\beta (\alpha )}(X;q,t),
\end{align}
where $E_{\mu}^{\sigma}(X;q,t)$ is the \emph{permuted basement Macdonald polynomial} introduced in \cite{Fer11} and further studied in \cite{Ale16}. We will define $G_{\gamma}$ combinatorially in the next section. Note that $E_{\mu}^{\sigma}$ is a polynomial in $k$ variables, where $k$ is the number of parts of $\mu$, so we actually mean $E_{\mu}^{\sigma}(X;q,t)=E_{\mu}^{\sigma}(x_1,\ldots,x_k;q,t)$, and $\sigma\in S_k$. 
\begin{remark}

It turns out that $E_{\inc(\alpha)}^{\beta (\alpha )}(X;0,t) = E_{\alpha}^{id}(X;0,t)$. Thus the quasisymmetric Hall--Littlewood polynomials $\mathcal{L}_{\alpha}(X;t)$, defined in \cite{HLMvW11} as
\[
\mathcal{L}_{\gamma}(X;t) = \sum_{\alpha : \alpha^+ = \gamma} E_{\alpha}^{id}(X;0,t),
\]
coincide with $G_{\gamma}(X;0,t)$.
\end{remark}

\subsection{Tableaux formula for $E_{\mu}^{\sigma}(X;q,t)$}

The polynomial $E_{\mu}^{\sigma}(X;q,t)$ has a combinatorial description in the form of a tableaux formula \cite{HHL05}. We review the relevant statistics for general compositions, though we will primarily focus on the case where the parts of $\mu$ are arranged in weakly increasing order.

\begin{comment}
%This is the old version with flipped arms and triples.
%
%
Define $\dg(\mu)$, the diagram of $\mu$, to be the composition shape in French notation with $\mu_i$ boxes in column $i$ from left to right. The rows are labeled from bottom to top, beginning with row 1; $(r,c)$ are the coordinates of a cell  in row $r$ and column $c$, and for a filling $T$ of $\dg(\alpha)$, $T(u)$ denotes the content of the cell $u\in\dg(\alpha)$, and a ``zeroth'' row, called the \emph{basement}, is appended to the bottom of $\dg(\mu)$. A tableau, or a filling, is a map $T:\dg(\mu) \rightarrow [1,k]$ where $k$ is the number of parts of $\mu$. The cells of the basement are filled from left to right with the elements of permutation $\sigma$. For example, $\mu=(4,3,3,1,0,0)$ for the tableau below, and it has basement $\sigma=(1,6,4,3,5,2)$. If $T$ is a filling of $\dg(\mu)$, we denote by $T(u)$ the entry in cell $u\in\dg(\mu)$. We note that $\dg(\mu)$ does not include the cells in the basement.

%\begin{center}
%\begin{tikzpicture}[scale=.5]
%\cell10{}
%\cell22{} \cell21{$$} \cell20{$$}  
%\cell32{$$}\cell31{$$}\cell30{} 
%\cell43{$3$} \cell42{$4$}\cell41{$6$}  \cell40{$1$} 
%\cell50{$1$} \cell51{$6$} \cell52{$4$}\cell53{$3$} \cell54{$5$} \cell55{$2$}  

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

Two cells are said to attack each other if they are in the same row, or if they are in adjacent rows with the one above to the northwest of the one below (i.e. in the configuration \begin{tikzpicture}[scale=0.4]\cell10{\ } \cell22{\ }\end{tikzpicture}\ ). A filling $T$ is considered \emph{non-attacking} if $T(u)\neq T(v)$ for any pair of attacking cells $u,v$. The set of \emph{non-attacking fillings} of $\dg(\mu)$ is denoted by $\NAT(\mu)$, and the set of such fillings with basement $\sigma$ is denoted by $\NAT_{\sigma}(\mu)$.

For a cell $u\in\dg(\mu)$, we call $\leg(u)$ the number of cells above $u$ in the same column. We call $\arm(u)$ the number of cells to the left of $u$ in columns not higher than the column containing $u$, plus the number of cells to the right of $u$ in columns strictly shorter than the column containing $u$. See \cref{fig:armleg}. Denote by $\South(u)$ the cell directly below $u$ in the same column. For technical reasons, if $u$ is in the bottom-most row, set $\South(u)=0$. The set of descents of a filling of $\dg(\mu)$ is
\[
\Des(T) = \{u\in\dg(\mu)\ :\ T(u)>T(\South(u))\},
\]
and the \emph{major index} is
\[
\maj(T) = \sum_{u\in\Des(T)} \leg(u)+1.
\]

\begin{figure}
  \centering
\begin{tikzpicture}[scale=.5]
\cell31{$a$} \cell32{} \cell33{$a$} \cell34{$u$}  \cell37{$$}\cell39{$$}
\cell41{} \cell42{} \cell43{$$}\cell44{}  \cell46{$a$} \cell47{} \cell49{$a$} %\cell49{}
\cell12{} 
\cell21{} \cell22{} \cell24{$\ell$} \cell27{}
\cell51{$$} \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 $\arm(u)=4$.}
\label{fig:armleg}
\end{figure}

(Inversion) triples consist of a cell $x$, the cell $y=\South(x)$ directly below, and a third cell $z$ in the arm of $x$. If $z$ is in the same row as $x$, this is called a Type A triple (the configuration \begin{tikzpicture}[scale=0.4]\cell12{$x$}\cell22{$y$} \cell10{$z$}\end{tikzpicture} \ ), and if $z$ is in the same row as $y$, this is called a Type B triple (the configuration \begin{tikzpicture}[scale=0.4]\cell22{$z$} \cell10{$x$}\cell20{$y$}\end{tikzpicture} \ ). Coinversion triples consist of Type A triples where the entries are increasing in counterclockwise orientation, plus Type B triples where the entries are increasing in clockwise orientation. All such triples contribute to the statistic $\coinv(T)$.

\begin{remark}
Due to the non-attacking condition, $z \neq x$ and $z \neq y$. Thus either all three entries are distinct, or  $x=y\neq z$. In the later case, the triple is not a coinversion triple.
\end{remark}

Let us call $f_{\alpha} =  E_{\inc(\alpha)}^{\beta (\alpha)}$, let $\lambda=\dec(\alpha)$ be the partition obtained from sorting the parts of $\alpha$ in non-increasing order, and let $\sigma$ be the permutation of longest length such that $\sigma \circ \alpha=\lambda$. 

\end{comment}

For any weak composition $\alpha$, define $\dg(\alpha)$, the diagram of $\alpha$, to be the composition shape in French notation with $\alpha_i$ boxes in column $i$ from left to right. The rows are labeled from bottom to top starting with row 1, and a cell in row $r$ and column $c$ is denoted by coordinates $(r,c)\in\dg(\alpha)$. Define $\widehat{\dg}(\alpha)$ to be the set of cells in $\dg(\alpha)$ not contained in the bottom row.  If $T$ is a filling of $\dg(\alpha)$, the entry in a cell $u\in\dg(\alpha)$ is denoted by $T(u)$. Let $x^T=\prod_{u \in \dg(\alpha)} x_{T(u)}$ be the monomial encoding the content of $T$.

%\begin{center}
%\begin{tikzpicture}[scale=.5]
%\cell10{}
%\cell22{} \cell21{$$} \cell20{$$}  
%\cell32{$$}\cell31{$$}\cell30{} 
%\cell43{$3$} \cell42{$4$}\cell41{$6$}  \cell40{$1$} 
%\cell50{$1$} \cell51{$6$} \cell52{$4$}\cell53{$3$} \cell54{$5$} \cell55{$2$}  
%
%\node at (-2.5,-.5) {row 4};
%\node at (-2.5,-1.5) {row 3};
%\node at (-2.5,-2.5) {row 2};
%\node at (-2.5,-3.5) {row 1};
%\node at (-2.5,-4.5) {row 0};
%\end{tikzpicture} 
%\end{center}

The \emph{reading order} of a diagram is the total order given by reading the entries along the rows from top to bottom, and from left to right within each row. Two cells are said to \emph{attack} each other if they are in the same row, or if they are in adjacent rows where the one above is strictly northeast of the one below. %, in the configuration
%\[\begin{tikzpicture}[scale=0.4]\cell12{\ } \cell20{\ }\end{tikzpicture}.\] 
A filling $T$ is considered \emph{non-attacking} if $T(u)\neq T(v)$ for any pair of attacking cells $u,v$. %The set of \emph{non-attacking fillings} of $\dg(\alpha)$ is denoted by $\NAT(\alpha)$.
%, and the set of such fillings with basement $\sigma$ is denoted by $\NAT_{\sigma}(\mu)$.

For a cell $u\in\dg(\alpha)$, we call $\leg(u)$ the number of cells above $u$ in the same column. We call $\arm(u)$ the number of cells to the right of $u$ in columns whose height does not exceed the height of the column containing $u$, plus the number of cells to the left of $u$ in columns of height strictly smaller than the height of the column containing $u$. More precisely, let $u=(r,i)$. Then
\begin{align*}
\arm(u) =& |\{(r,j)\in\dg(\alpha):\ j>i, \alpha_j\leq \alpha_i\}|+|\{(r-1,j)\in\dg(\alpha): j<i, \alpha_j< \alpha_i\}|
\end{align*} 
See \cref{fig:armleg}. Denote by $\South(u)$ the cell directly below $u$ in the same column. The set of descents of a filling of $\dg(\alpha)$ is
\[
\Des(T) = \{u\in\widehat{\dg}(\alpha)\ :\ T(u)>T(\South(u))\},
\]
and the \emph{major index} is
\[
\maj(T) = \sum_{u\in\Des(T)} \leg(u)+1.
\]

\begin{figure}
  \centering
\begin{tikzpicture}[scale=.5]
\cell39{$a$} \cell38{} \cell37{$a$} \cell36{$u$}  \cell33{$$}\cell31{$$}
\cell49{} \cell48{} \cell47{$$}\cell46{}  \cell44{$a$} \cell43{} \cell41{$a$} %\cell49{}
\cell18{} 
\cell29{} \cell28{} \cell26{$\ell$} \cell23{}
\cell59{$$} \cell58{$$}  \cell57{$$} \cell56{$$}  \cell55{$$}  \cell54{$$}  \cell53{$$}  \cell52{$$}  \cell51{$$} 

\node at (-2.5,-.5) {row 5};
\node at (-2.5,-1.5) {row 4};
\node at (-2.5,-2.5) {row 3};
\node at (-2.5,-3.5) {row 2};
\node at (-2.5,-4.5) {row 1};
\end{tikzpicture} 
\caption{The diagram of the composition $(3,1,4,2,1,4,3,5,4)$ and the cells in the leg and the arm of the cell $u=(3,6)$. Here $\leg(u)=1$ and $\arm(u)=4$.}
\label{fig:armleg}
\end{figure}

\emph{Triples} consist of a cell $x$, the cell $y=\South(x)$ directly below, and a third cell $z$ in the arm of $x$. If $z$ is in the same row as $x$, this is called a \emph{type A triple}, and if $z$ is in the same row as $y$, this is called a \emph{type B triple}, as shown:

\begin{align*}
\begin{tabular}{b{3.5em}m{5em}b{3.5em}m{3.5em}}
\text{Type A: } & \begin{tikzpicture}[scale=0.4]\cell10{$x$}\cell20{$y$} \cell12{$z$}\end{tikzpicture}&
\text{ Type B: }& \begin{tikzpicture}[scale=0.4]\cell20{$z$} \cell12{$x$}\cell22{$y$}\end{tikzpicture}
\end{tabular}
\end{align*}

Coinversion triples consist of type A triples where the entries are increasing in clockwise orientation, plus type B triples where the entries are increasing in counterclockwise orientation. The $\coinv(T)$ statistic is defined as the total number of all such triples. 

%\begin{defn}
Let $\gamma$ be a strong composition, and let $\sigma=\beta(\gamma)$ be the longest permutation %of longest length 
such that $\sigma \circ \gamma=\inc(\gamma)$. Define $\NAT(\gamma)$ to be the set of non-attacking fillings of $\dg(\inc(\gamma))$ such that the entries of the first row are order-equivalent to $\sigma$ when read in reading order. 

\begin{example}
Let $\alpha=(0,4,0,3,1,0,0,3)$. Then $\inc(\alpha)=(0,0,0,0,1,3,3,4)$,  $\alpha^+=(4,3,1,3)$, and $\beta(\alpha)=(7,6,3,1,5,8,4,2)$. The NAT associated to $\alpha$ are fillings of $\dg(\inc(\alpha^+))$ with the bottom row equal to $(5,8,4,2)$: the last $\ell$ entries of $\beta(\alpha)$, where $\ell=\ell(\alpha^+)=4$. %(3,4,2,1)$. 
Notice that $(5,8,4,2)$, is order-equivalent to $(3,4,2,1)$, and $(3,4,2,1)=\beta(\alpha^+)$. Thus in particular, all tableaux associated to $\alpha$ also belong to $\NAT(\alpha^+)$, such as the one below.
\begin{center}
%\begin{tikzpicture}[scale=.5]
%\cell13{}
%\cell21{} \cell22{$$} \cell23{$$}  
%\cell31{$$}\cell32{$$}\cell33{} 
%\cell40{$3$} \cell41{$4$}\cell42{$2$}  \cell43{$1$} 
%\end{tikzpicture} \qquad
\begin{tikzpicture}[scale=.5]
\cell13{5}
\cell21{2} \cell22{$7$} \cell23{$5$}  
\cell31{$4$}\cell32{$1$}\cell33{2} 
\cell40{$5$} \cell41{$8$}\cell42{$4$}  \cell43{$2$} 
\node at (6.5,-2) {$\in \NAT((4,3,1,3))$};
\end{tikzpicture} %\qquad
%\begin{tikzpicture}[scale=.5]
%\cell13{}
%\cell21{} \cell22{$$} \cell23{$$}  
%\cell31{$$}\cell32{$$}\cell33{} 
%\cell40{$7$} \cell41{$11$}\cell42{$3$}  \cell43{$1$} 
%\end{tikzpicture} \qquad
%\begin{tikzpicture}[scale=.5]
%\cell13{}
%\cell21{} \cell22{$$} \cell23{$$}  
%\cell31{$$}\cell32{$$}\cell33{} 
%\cell40{$6$} \cell41{$8$}\cell42{$5$}  \cell43{$2$} 
%\end{tikzpicture} 
\end{center}
\end{example}

By comparing with \cite{Ale16}, we obtain the combinatorial formula for $E_{\inc(\alpha)}^{\beta(\alpha)}(X;q,t)$, where $\alpha$ is a weak composition:
%
\begin{equation}\label{eq:E}
E_{\inc(\alpha)}^{\beta (\alpha)} (X; q,t) = \sum_{\substack{T\in\NAT(\alpha^+)\\T\ \text{has bottom row } \pi}} %q^{\maj(T)}t^{\coinv(T)}
\wt(T)x^T,
%\prod_{\substack{u \in \widehat{\dg}(\alpha^+)\\T(u) \neq T(\South(u))}} \frac{(1-t)}{(1-q^{\leg(u)+1}t^{\arm(u)+1})},
\end{equation}
where $\pi$ is the last $\ell$ entries of $\beta(\alpha)$, for $\ell=\ell(\alpha^+)$.
%$\sigma=(\sigma_{m-\ell+1},\ldots,\sigma_{m-1},\sigma_m)$ with $m$ being the total number of parts in $\alpha$, $\ell$ the number of nonzero parts, and $(\sigma_1,\ldots,\sigma_m)=\beta(\alpha)$. \todo{is this formula ok as written?}
Here, the weight of a (nonstandard) filling $T$ is
\begin{align}
\wt(T)= q^{\maj(T)}t^{\coinv(T)} \prod_{\substack{u \in \widehat{\dg}(\alpha^+)\\T(u) \neq T(\South(u))}} \frac{(1-t)}{(1-q^{\leg(u)+1}t^{\arm(u)+1})}
\end{align}


\begin{remark}
We have given the tableaux formula for $E_{\mu}^{\sigma}$ where the parts of $\mu$ are weakly increasing. A general formula exists (see \cite{Ale16} for details) for an arbitrary composition $\mu$ and a permutation $\sigma$ by keeping track of the ``basement'' of a filling. Comparing definitions, it follows that for any composition $\alpha$, the basement of a filling of $\dg(\inc(\alpha))$ can be recovered uniquely from the bottom row of the filling.
% It turns out that when $\lambda$ is a partition, for a filling of $\dg(\lambda)$ to be non-attacking, each entry in the first row must match the entry directly below it in the basement. For example, we have filled the first row of the tableau above with the unique possible filling of that row, for the fixed basement $\sigma=(1,6,4,3,5,2)$. Moreover, since $\beta(\alpha)$ for a composition $\alpha$ is of longest length, the basement entries in columns of equal heights must be strictly decreasing from left to right. Thus the basement can be reconstructed uniquely from the first row of a filling of $\dg(\lambda)$. Since we will only be working with tableaux corresponding to partitions, we will omit the basement from now on. \todo{replace this remark with one about combinatorial defn of $E_{\inc(\alpha)}^{\sigma}$}
\end{remark}




\subsection{Standard, packed, and non attacking fillings}

A \emph{packed} filling is one that uses every integer from the set $\{1,\ldots,m\}$ for some $m$. %(not including the basement). 
%For example, the filling \begin{tikzpicture}[scale=0.4]\cell01{$2$} \cell11{$2$}\cell10{$1$}\end{tikzpicture} is packed, but \begin{tikzpicture}[scale=0.4]\cell01{$3$} \cell11{$3$}\cell10{$1$}\end{tikzpicture} is not. 
%However, the latter 
Any filling compresses to a packed filling by shifting the alphabet of values in the filling down as necessary: given a set $\{s_1,\ldots,s_k\}$ with $s_1<\cdots<s_k$, the entries $s_i$ become $i$.

It is convenient to work with packed fillings in the context of quasisymmetric functions. We consider every packed filling $T$ to be the representative of the family of fillings which compress to $T$. 
\begin{lemma}
Suppose $T'\in\NAT(\gamma)$ compresses to a packed filling $T\in\NAT(\gamma)$. Then $\coinv(T')=\coinv(T)$ and $\maj(T')=\maj(T)$.
\end{lemma}

The proof of the above lemma follows from the fact that the relative order of entries is preserved by compression. Moreover,
\[
\sum_{T'} x^{T'} = M_T,
\]
the sum being over all fillings $T'$ that compress to the packed filling $T$, and $M_T$ is the monomial quasisymmetric function corresponding to the content of $T$. Thus the $q,t$-generating function of the family of fillings that compress to the packed representative $T$ is the weight of $T$ times $M_T$. 
Hence, we may work with the finite set of packed fillings to represent all possible fillings.



From \eqref{def: Mac} and \eqref{eq:F}, we thus obtain
\begin{equation}\label{eq:G}
G_{\gamma}(X;q,t) %&= \sum_{\alpha: \alpha^+=\gamma} E_{\inc(\alpha)}^{\beta (\alpha)} \nonumber\\
= \sum_{\substack{T\in\NAT(\gamma)\\T\ \text{packed}}} q^{\maj(T)}t^{\coinv(T)}M_T \prod_{\substack{u \in \widehat{\dg}(\gamma)\\T(u) \neq T(\South(u))}} \frac{(1-t)}{(1-q^{\leg(u)+1}t^{\arm(u)+1})}.
\end{equation}


\begin{example}
For $\gamma=(1,2)$, all the packed nonattacking fillings in $\NAT(\gamma)$ are shown below with their weights, to obtain
\[
G_{(1,2)} = M_{\{1\}}+\frac{(1-t)(1+t+qt)}{1-qt^2}M_{\{1,1\}}.
\]
\begin{center}
\begin{tikzpicture}[scale=.5]
\node at (-4,0) {$\gamma=(1,2)$:};
\cell01{2}
\cell10{1} \cell11{2}
\node at (0,-2) {$M_{\{1\}}$};
\cell06{3}
\cell15{1} \cell16{2}
\node at (5,-2) {$\frac{qt(1-t)}{1-qt^2}M_{\{1,1\}}$};
\cell0{11}{2}
\cell1{10}{1} \cell1{11}{3}
\node at (10,-2) {$\frac{(1-t)}{1-qt^2}M_{\{1,1\}}$};
\cell0{16}{1}
\cell1{15}{2} \cell1{16}{3}
\node at (15,-2) {$\frac{t(1-t)}{1-qt^2}M_{\{1,1\}}$};
\end{tikzpicture}
\end{center}
\end{example}

\emph{Standard fillings} (or standard tableaux), denoted by $\ST(\gamma)$, are fillings of $\dg(\inc(\gamma))$ such that every element in the set $\{1,\ldots,n\}$ appears exactly once, where $n=|\gamma|$. Thus there is a bijection $\tau:\dg(\inc(\gamma))\rightarrow\{1,\ldots,n\}$ between cells of $\dg(\inc(\gamma))$ and the entries $\{1,\ldots,n\}$, and so we can slightly abuse notation and refer to both a cell and its entry when we work with standard tableaux. 

Define the standardization map $\std \colon \NAT(\gamma) \rightarrow \ST(\gamma)$ as follows. For $T\in\NAT(\gamma)$, let $\tau=\std(T)$ be the unique standard filling in $\ST(\gamma)$ that preserves the relative order of the original tableau, and where the reading order is used to break ties. It is straightforward to check that if $\tau$ is the standardization of $T$, then $\coinv(T)=\coinv(\tau)$ and $\maj(T)=\maj(\tau)$. See \cref{example:std} for the standardization $\std(T)$ of $T\in\NAT((1,4,3))$. 

Let $T\in\NAT(\gamma)$ with standardization $\tau=\std(T)$, and $n=|\gamma|$. Define the \emph{reading word} of $T$ to be the sequence of entries of $T$ listed in reading order, denoted by $\rw(T)$. The reading word of $\tau$ is thus a permutation of $\{1,\ldots,n\}$. Define $\ID(\tau)$ to be the \emph{inverse descent set}, where  $i\in \ID(\tau)$ if $i+1$ precedes $i$ in $\rw(\tau)$.

\begin{example}\label{example:std}
We show $T\in\NAT((1,4,3))$ and the corresponding standardization $\tau=\std(T)$. We have 
%$a=T\circ\tau^{-1}=(1,1,2,2,3,3,4,5)$, 
$\rw(\tau)=(3,5,1,8,6,2,7,4)$, and $\ID(\tau)=\{2,4,7\}$. %We check that the indices $i$ for which $a(i)=a(i+1)$ are $\{1,3,5\}$, and indeed those are not contained in $\ID(\tau)$.
\begin{center}
\begin{tikzpicture}[scale=0.5]
\node at (-2,-1) {$T=$};
\cell022
\cell121\cell113
\cell223\cell215
\cell322\cell314\cell301
\end{tikzpicture} 
\qquad
\begin{tikzpicture}[scale=0.5]
\node at (-3,-1) {$\tau=\std(T)=$};
\cell023
\cell121\cell115
\cell226\cell218
\cell324\cell317\cell302
\end{tikzpicture} 
\end{center}
\end{example}




For $\tau\in\ST(\gamma)$, define %the \emph{index}
 $V(\tau)\subseteq [n-1]$ to be the set of entries such that $i\in V(\tau)$ if $i\in\ID(\tau)$ or if $i$ and $i+1$ are in cells that attack each other. Define $W(\tau) = \{i\in \tau\ :\ \South(i)=i+1\}$ to be the set of entries $i$ with $i+1$ directly below. Note that $V(\tau) \cap W(\tau) = \emptyset$.

Given a standard filling $\tau$ with $n$ cells, the cells labelled from 1 to $n-1$ are partitioned into three blocks:
\begin{itemize}%[wide,itemsep=0pt]
\item The cells with entries in $V(\tau)$, namely those cells where $i\in\ID(\tau)$ 
{\bf OR} $i$ and $i+1$ are in attacking cells.
\item The cells with entries in $W(\tau)$, namely those cells where $i+1$ is directly below $i$.
\item The rest of the cells with entries in $[n-1]\backslash (V(\tau)\cup W(\tau))$.
\end{itemize}


Let $\gamma \models n$. We consider the pre-image in $\NAT(\gamma)$ of standard fillings $\tau\in\ST(\gamma)$. For a choice of $V(\tau)\subseteq S\subseteq [n-1]$, define a \emph{destandardization map} $\delta_S(\tau)\ :\  \dg(\gamma) \rightarrow \mathbb{Z}$ as follows. Let $\alpha$ be the (strong) composition corresponding to the set $S$. Let $w$ be the word containing the content associated to $\alpha$ in weakly decreasing order, given by $w=(1^{\alpha_1},2^{\alpha_2},\ldots,k^{\alpha_k})$ where $\alpha$ has $k$ parts. Define $\delta_S(\tau):= w\circ \tau$ to be the unique filling of $\dg(\gamma)$ with content $\alpha$ that standardizes to $\tau$. 

\begin{example}
Consider the standard tableau $\tau$ from \cref{example:std}. $\ID(\tau)=\{2,4,7\}$, and the set of indices $i$ such that $i$ and $i+1$ are in cells that attack each other is $\{6\}$, so $V(\tau)=\{2,4,6,7\}$. Thus, $S$ can be any subset of $[7]$ containing $V(\tau)$. We show some examples of $\delta_S$ for various choices of $S$:
\begin{center}
\begin{tikzpicture}[scale=0.5]
\node at (-3,-1) {$\delta_{\left\{\substack{1,2,3,\\4,5,6,7}\right\}}(\tau)=$};
\cell023
\cell121\cell115
\cell226\cell218
\cell324\cell317\cell302
\end{tikzpicture} 
\quad 
\begin{tikzpicture}[scale=0.5]
\node at (-2.9,-1) {$\delta_{\left\{\substack{1,2,4,\\5,6, 7}\right\}}(\tau)=$};
\cell023
\cell121\cell114
\cell225\cell217
\cell323\cell316\cell302
\end{tikzpicture} 
%\ \ 
%\begin{tikzpicture}[scale=0.5]
%\node at (-2,-1) {$\delta_{\left\{\substack{2,3,\\4,6}\right\}}(\tau)=$};
%\cell022
%\cell121\cell114
%\cell224\cell215
%\cell323\cell315\cell301
%\end{tikzpicture} 
\quad 
\begin{tikzpicture}[scale=0.5]
\node at (-2.8,-1) {$\delta_{\left\{ \substack{2,4, \\ 6, 7}\right\}}  (\tau)=$};
\cell022
\cell121\cell113
\cell223\cell215
\cell322\cell314\cell301
\end{tikzpicture} 
\end{center}

%\begin{center}
%\begin{tikzpicture}[scale=0.5]
%\node at (-1.8,-1) {$\delta_{\{1,3\}}(\tau)=$};
%\cell023
%\cell122\cell115
%\cell225\cell216
%\cell324\cell316\cell301
%\end{tikzpicture} 
%\quad
%\begin{tikzpicture}[scale=0.5]
%\node at (-1.8,-1) {$\delta_{\{3,5\}}(\tau)=$};
%\cell022
%\cell121\cell114
%\cell225\cell216
%\cell323\cell316\cell301
%\end{tikzpicture} 
%\quad
%\begin{tikzpicture}[scale=0.5]
%\node at (-1.6,-1) {$\delta_{\{1\}}(\tau)=$};
%\cell023
%\cell122\cell114
%\cell224\cell215
%\cell323\cell315\cell301
%\end{tikzpicture} 
%\quad
%\begin{tikzpicture}[scale=0.5]
%\node at (-1.6,-1) {$\delta_{\{5\}}(\tau)=$};
%\cell022
%\cell121\cell113
%\cell224\cell215
%\cell322\cell315\cell301
%\end{tikzpicture} 
%\end{center}

\end{example}

%The following Lemmas contain some observations that will be essential to our arguments.
%
%\begin{lemma}\label{lem:std}
%Let $\gamma\models n$, and let $\tau\in\ST(\gamma)$. For a set $S\subseteq [n-1]$ such that $V(\tau)\subseteq S$, the following conditions hold: 
%\begin{enumerate}[i.]
%\item $\delta_S(\tau)\in \NAT(\gamma)$.
%\item $\std(\delta_S(\tau))=\tau$.
%\item Let $S$ be the unique set such that $T=\delta_S(\tau)$. Then $\wt(T) =  \wt(\delta_{S}(\std(T)))$.
%\end{enumerate}
%\end{lemma}
%
%In fact, the destandardization map splits $\NAT(\gamma)$ into disjoint components $\{T\in \NAT(\gamma)\ :\ \std(T)=\tau\}$ indexed by $\tau\in\ST(\gamma)$, due to the following lemma.
%
%\begin{lemma}\label{lem:delta}
%Let $\gamma\models n$, let $T\in\NAT(\gamma)$ be a packed filling, and set $\tau=\std(T)$. There exists a unique set $V(\tau)\subseteq S \subseteq [n-1]$ such that $T=\delta_S(\tau)$. 
%Furthermore,
%\begin{align*}
%\sum_{T\ :\ \std(T) =\tau} x^T 
%=\sum_{V(\tau) \subseteq S \subseteq [n-1]} \sum_{T=\delta_S(\tau)} M_T
%= \sum_{V(\tau) \subseteq S \subseteq [n-1]} M_{S}.
%\end{align*} 
%\end{lemma}
%
%Thus for a strong composition $\gamma$, the set of packed fillings contained in $\NAT(\gamma)$ is
%\[
%\bigsqcup_{\tau\in\ST(\gamma)} \bigcup_{V(\tau)\subseteq S\subseteq [n-1]} \delta_S(\tau).
%\]

%Now we show that the weight we defined on standard tableaux is related to the weight of the destandardized $\NAT$ by the following.



%\begin{lemma}\label{lem:wt}
%Let $\tau\in\ST(\gamma)$, $S$ such that $V(\tau)\subseteq S\subseteq [n-1]$, and let $U=S\cap W(\tau)$. Then 
%\[
%\wt(\delta_S(\tau))=\wt_{\ST}(\tau)(1-t)^{|S\cap W(\tau)|}=t^{\coinv(\tau)}(1-t)^{h(\gamma)-|W(\tau)\backslash S|}.
%\]
%\end{lemma}


%\begin{proof}
%The coinversions are preserved by standardization, so the only difference between the power of $(1-t)$ in $\wt(\delta_S(\tau))$ and $\wt_{\ST_0}(\tau)$ is the number of cells $u$ in $\delta_S(\tau)$ such that $\South(u)=u$ vs. the number of entries $i$ in $\tau$ such that $\South(i)=i+1$. But by our definitions, the first quantity is precisely given by $h(\gamma)-(|W(\tau)|-|U|)$ and the second is $h(\gamma)-|W(\tau)|$. 
%\end{proof}

The following lemma gives the weight of a destandardized filling in terms of its standardization.

\begin{lemma}\label{lem:wtq}
Let $\gamma\models n$, $\tau\in\ST(\gamma)$, and $S$ such that $V(\tau)\subseteq S\subseteq [n-1]$. Then 
\begin{align*}
\wt(\delta_S(\tau)) = &\ t^{\coinv(\tau)} q^{\maj(\tau)} \prod_{\substack{u \in \widehat{\dg}(\gamma)\\ u\not\in W(\tau)}} \frac{1-t}{1-q^{\leg(u)+1}t^{\arm(u)+1}}
 %\\ &
   \prod_{u \in S\cap W(\tau)} \frac{1-t}{1-q^{\leg(u)+1}t^{\arm(u)+1}}.
\end{align*}
\end{lemma}
%\begin{proof}
%We are given $V(\tau)\subseteq S\subseteq [n-1]$, and let $T=\delta_S(\tau)$. Recall that $W(\tau)$ contains the entries $\{i\in\tau: \South(i)=i+1\}$.  Ensuring the products are correct is a matter of tracking the cells $\{u\in\widehat{\dg}(T) : T(\South(u))=T(u)\}$.  All such cells must correspond to cells of $W(\tau)$.  %ll which correspond to the only possible choices of cells mapped by $\delta_S$ to have the latter property. 
%In particular, by our construction, $\{u\in\widehat{\dg}(T):T(u)=T(\South(u))\} = W(\tau)\backslash S$.  All other cells contribute a factor of $\frac{1-t}{1-q^{\leg(u)}t^{\arm(u)}}$.
%Since the coinversions and major index are preserved by standardization, the lemma follows.
%\end{proof}

For a strong composition $\gamma \models n$ where $\ell(\gamma)$ is the number of parts, define $h(\gamma)=n-\ell(\gamma)$ to be the number of cells in $\dg(\gamma)$ without its bottom row. Note that $h(\gamma)$ is the number of cells in $\widehat{\dg}(\gamma)$. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proof of \cref{thm:q}}\label{sec:packed}

We will start with a proof for the $q=0$ specialization of \cref{thm:q} to develop the main ideas of the proof. The proof for the general $q$ case follows the same strategy.
%The ideas of the proof are about the same; the equations become much longer.



\subsection{The $q=0$ case}\label{sec:q0}

We assume $q=0$ throughout this section. Let $\gamma$ be a strong composition. When we compute compute $G_{\gamma}(X;0,t)$, the only surviving tableaux in \eqref{eq:G} are those with an empty descent set, which means the entries must be non-increasing as we read the columns from bottom to top. We denote the subsets of $\NAT(\gamma)$ and $\ST(\gamma)$ that have nonzero weight at $q=0$ by $\NAT_0(\gamma)$ and $\ST_0(\gamma)$, respectively.  We will prove the following.
\begin{equation}\label{eq:HL}
G_{\gamma}(X;0,t) = \sum_{\tau\in\ST_0(\gamma)} t^{\coinv(\tau)}(1-t)^{h(\gamma)-|W(\tau)|} \sum_{U\subseteq W(\tau)} (-t)^{|U|} F_{V(\tau)\cup U}.
\end{equation}

%Let $\gamma$ be a strong composition. 
Observe the following. The denominator in the product of \eqref{eq:G} vanishes, and the weight of each $T\in\NAT_0(\gamma)$ becomes
\[
\wt(T)=t^{\coinv(T)}(1-t)^{|\{u \in \widehat{\dg}(\lambda)\ :\ T(\South(u))\neq T(u)\}|}.
\]
Moreover, for $\tau\in\ST_0(\gamma)$, \cref{lem:wtq} specializes to
\begin{equation}\label{eq:wt}
\wt(\delta_S(\tau))=t^{\coinv(\tau)}(1-t)^{h(\gamma)-|W(\tau)\backslash S|}.
\end{equation}
%For standard tableaux, we similarly define the weight $\wt_{\ST_0}(\tau)$ of $\tau\in\ST_0(\gamma)$ to be:
%\[
%\wt_{\ST_0}(\tau)=t^{\coinv(\tau)}(1-t)^{|\{i \in \tau\ :\ \South(i)\neq i+1\}|}
%\]


 

We are now ready to prove \cref{thm:q} at $q=0$. %the main result for $q=0$.

\begin{proof}[Proof of \eqref{eq:HL}]
From the definition, we have
\begin{align}
G_{\gamma}(X;0,t) &= \sum_{T\in\NAT_0(\gamma)} \wt(T)x^T \nonumber\\
&=\sum_{\tau\in\ST_0(\gamma)} \sum_{V(\tau)\subseteq S \subseteq [n-1]} \wt(\delta_S(\tau))M_{S}\nonumber\\
&=\sum_{\tau\in\ST_0(\gamma)} \sum_{V(\tau)\subseteq S \subseteq [n-1]} t^{\coinv(\tau)}(1-t)^{h(\gamma)-|W(\tau)\backslash S|} M_{S}\nonumber\\
&=\sum_{\tau\in\ST_0(\gamma)}   t^{\coinv(\tau)}(1-t)^{h(\gamma)-|W(\tau)|}\sum_{V(\tau)\subseteq S \subseteq [n-1]}  
(1-t)^{|S\cap W(\tau)|}M_{S} \label{eq:G2}
\end{align}
where the third line is by \eqref{eq:wt}. We then reformulate the second summation:
\begin{align*}\sum_{V(\tau)\subseteq S \subseteq [n-1]}  
(1-t)^{|S\cap W(\tau)|}M_{S} 
&=\sum_{W\subseteq W(\tau)} (1-t)^{|W|} \sum_{\substack{V(\tau)\subseteq S \subseteq [n-1]\\S\cap W(\tau)=W}} M_{S}
\end{align*}
By the binomial theorem, $(1-t)^{|W|} = \sum_{U \subseteq W} (-t)^{|U|}$. 
%\begin{equation}\label{eq:W}
%(1-t)^{|W|} = \sum_{U \subseteq W} (-t)^{|U|}.
%\end{equation}
Plugging in gives
\begin{align*}
\sum_{W\subseteq W(\tau)}\sum_{U \subseteq W} (-t)^{|U|} \sum_{\substack{V(\tau)\subseteq S \subseteq [n-1]\\S\cap W(\tau)=W}} M_{S}
&=\sum_{U \subseteq W(\tau)} (-t)^{|U|} \sum_{W\supseteq U}\sum_{\substack{V(\tau)\subseteq S \subseteq [n-1]\\S\cap W(\tau)=W}} M_{S}\\
&=\sum_{U \subseteq W(\tau)} (-t)^{|U|} \sum_{V(\tau)\cup U\subseteq S \subseteq [n-1]} M_{S}\\
&=\sum_{U \subseteq W(\tau)} (-t)^{|U|} F_{V(\tau)\cup U},
\end{align*}
%
%\begin{align*}
%G_{\gamma}(X;0,t)& =\sum_{\tau\in\ST_0(\gamma)} t^{\coinv(\tau)}(1-t)^{h(\gamma)-|W(\tau)|}\sum_{V(\tau)\cup W(\tau) \subseteq S \subseteq [n-1]} \sum_{W\subseteq W(\tau)} M_{S\cup W} \sum_{U \subseteq W} (-t)^{|U|} \\
%&=\sum_{\tau\in\ST_0(\gamma)} t^{\coinv(\tau)}(1-t)^{h(\gamma)-|W(\tau)|} \sum_{W\subseteq W(\tau)}  (-t)^{|W|} \sum_{W\cup V(\tau) \subseteq S\subseteq [n-1]} M_{S} \\
%&=\sum_{\tau\in\ST_0(\gamma)} t^{\coinv(\tau)}(1-t)^{h(\gamma)-W(\tau)} \sum_{W \subseteq W(\tau)} (-t)^{|W|} 
%F_{W\cup V(\tau)}.
%\end{align*}
which completes the proof.
\end{proof}


\subsection{The general $q$ case}
Let $\gamma$ be a strong composition of $n$. Recall that the weight of a tableau $T\in\NAT(\gamma)$ is% given by:
\[
\wt(T) = t^{\coinv(T)} q^{\maj(T)} \prod_{\substack{u \in \widehat{\dg}(\gamma)\\T(\South(u))\neq T(u)}} \frac{1-t}{1-q^{\leg(u)+1}t^{\arm(u)+1}}.
\]
%Similarly, we defined $\ST(\gamma)$ to be the set of standardizations of tableaux in $\NAT(\gamma)$.


%We use the same destandardization map as in the $q=0$ case. Now, for $\tau\in \ST(\gamma)$, it is still the case that the only cells that have the potential to
\noindent As in the $q=0$ case, we split the set $\NAT(\gamma)$ via the destandardization map $\delta_S$ into disjoint components indexed by their representative standard fillings in $\ST(\gamma)$. For $\tau\in\ST(\gamma)$ and any $V(\tau)\subseteq S \subseteq [n-1]$, again, the only cells that will potentially change the weight of the destandardized tableau $\delta_S(\tau)$ are those in $W(\tau)$. This is because when an entry $i\in \tau$ has $i+1$ above it, it is possible for that pair to destandardize to the same value if $\delta_S(\tau)\circ \tau^{-1}(i)=\delta_S(\tau)\circ \tau^{-1}(i+1)$, changing the product in the weight function.


%To prove \cref{thm:q}, we 
We require the following lemma.

\begin{lemma}
Let $W$ be any subset of the cells of $\dg(\lambda)$. Then
\[
\prod_{u \in W} \frac{1-t}{1-q^{\leg(u)+1}t^{\arm(u)+1}} =\sum_{U \subseteq W} (-t)^{|U|} \left( \prod_{u \in U} \frac{1-q^{\leg(u)+1}t^{\arm(u)}}{1-q^{\leg(u)+1}t^{\arm(u)+1}} \right).
\]
\label{bino}
\end{lemma}
%\begin{proof}
%First we note that
%\[
%\frac{1-t}{1-q^{\leg(u)+1}t^{\arm(u)+1}} = 1+ \frac{-t+q^{\leg(u)+1}t^{\arm(u)+1}}{1-q^{\leg(u)+1}t^{\arm(u)+1}} = 1+ \frac{-t(1-q^{\leg(u)+1}t^{\arm(u)})}{1-q^{\leg(u)+1}t^{\arm(u)+1}}
%\] 
%Then we apply the binomial theorem. 
%Given a set $S$ and a function $f:S\rightarrow \mathbb{Q}(q,t)$, the binomial theorem states
%\[
%\prod_{s\in S}(1+f(s))=\sum_{U\subseteq S}\left(\prod_{u\in U}f(u)\right).
%\]
%We apply this with $f(u)=\frac{-t\left(1-q^{\leg(u)+1}t^{\arm(u)}\right)}{1-q^{\leg(u)+1}t^{\arm(u)+1}}$.
%\end{proof}



We can now prove the main result.
\begin{proof}[Proof of \cref{main}]
The proof is now completely analogous to the $q=0$ case with addition of \cref{bino} in place of the binomial theorem.
%
%By \cref{lem:std,lem:delta} we have
%\begin{align*}
%G_{\gamma}(X;q,t) &= \sum_{T\in\NAT(\gamma)} \wt(T)x^T\\
%&=\sum_{\tau\in\ST(\gamma)}\sum_{V(\tau)\subseteq S \subseteq [n-1]}  \wt(\delta_S(\tau)) M_{S}.\\
%\end{align*}
%
%\noindent
%Applying \cref{lem:wtq},
%\begin{align*}
%G_{\gamma}(X;q,t) =& \sum_{\tau\in\ST(\gamma)} t^{\coinv(\tau)}q^{\maj(\tau)}\left( \prod_{\substack{u \in \widehat{\dg}(\gamma)\\u\not\in W(\tau)}} \frac{1-t}{1-q^{\leg(u)+1}t^{\arm(u)+1}} \right)\\
%&
%\times \sum_{V(\tau)\subseteq S \subseteq [n-1]} \left( \prod_{u \in S\cap W(\tau)} \frac{1-t}{1-q^{\leg(u)+1}t^{\arm(u)+1}} \right) M_{S}\\
%=&\sum_{\tau\in\ST(\gamma)} t^{\coinv(\tau)}q^{\maj(\tau)}\left( \prod_{\substack{u \in \widehat{\dg}(\gamma)\\u\not\in W(\tau)}} \frac{1-t}{1-q^{\leg(u)+1}t^{\arm(u)+1}} \right)\\
%&\times \sum_{W\subseteq W(\tau)}\left( \left( \prod_{u \in W} \frac{1-t}{1-q^{\leg(u)+1}t^{\arm(u)+1}} \right)
%\sum_{\substack{V(\tau)\subseteq S \subseteq [n-1]\\S\cap W(\tau)=W}} M_{S}\right).
%\end{align*}
%
%\noindent
%By \cref{bino},
%\begin{align*}
%&\sum_{W\subseteq W(\tau)}\left( \left( \prod_{u \in W} \frac{1-t}{1-q^{\leg(u)+1}t^{\arm(u)+1}} \right)
%\sum_{\substack{V(\tau)\subseteq S \subseteq [n-1]\\S\cap W(\tau)=W}} M_{S}\right)\\
%&=\sum_{W\subseteq W(\tau)}\left(\sum_{U \subseteq W} (-t)^{|U|} \left( \prod_{u \in U} \frac{1-q^{\leg(u)+1}t^{\arm(u)}}{1-q^{\leg(u)+1}t^{\arm(u)+1}} \right)\sum_{\substack{V(\tau)\subseteq S \subseteq [n-1]\\S\cap W(\tau)=W}} M_{S}\right).
%\end{align*}
%Next, we reorder the summations, as
%\begin{align*}
%&=\sum_{U \subseteq W(\tau)} \left((-t)^{|U|} \left( \prod_{u \in U} \frac{1-q^{\leg(u)+1}t^{\arm(u)}}{1-q^{\leg(u)+1}t^{\arm(u)+1}} \right)\sum_{\substack {W\subseteq W(\tau)\\ W\supseteq U}}\sum_{\substack{V(\tau)\subseteq S \subseteq [n-1]\\S\cap W(\tau)=W}} M_{S}\right).
%\end{align*}
%Finally, we gather terms and apply \eqref{eq:F},
%\begin{align*}
%&=\sum_{U \subseteq W(\tau)} \left((-t)^{|U|} \left( \prod_{u \in U} \frac{1-q^{\leg(u)+1}t^{\arm(u)}}{1-q^{\leg(u)+1}t^{\arm(u)+1}} \right)\sum_{V(\tau)\cup U \subseteq S \subseteq [n-1]} M_{S}\right)\\
%&=\sum_{U \subseteq W(\tau)} (-t)^{|U|} \left( \prod_{u \in U} \frac{1-q^{\leg(u)+1}t^{\arm(u)}}{1-q^{\leg(u)+1}t^{\arm(u)+1}} \right)F_{V(\tau)\cup U}.
%\end{align*}
%The theorem follows.
\end{proof}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Further simplifications and specializations}\label{sec:HL}
%\subsection{The Hall--Littlewood case}

In this section, we further simplify the result for the Hall--Littlewood case ($q=0$) from \cref{sec:q0}. First, we introduce some notation. Let 
\begin{align*}
\ST_{1}(\gamma)= \{\tau \in \ST(\gamma)\colon i \in \Des(\tau) \implies \South(i) = i-1\}.
\end{align*}
That is, reading down columns, values can decrease by at most 1 per cell. 
We may send any element of $\tau' \in \ST_1(\gamma)$ to an element of $\tau \in \ST_0(\gamma)$ by sorting entries within their columns to become weakly decreasing from bottom to top. In this case the cells containing descents are sent to some $U\subset W(\tau)$, though the values in the cells may change. To make this sorting function invertible, we need to keep track of $U$. For any $U\subseteq W(\tau)$, consider the map $\iota_U$ that sends  $\tau \in \ST_0(\gamma)$ to $\tau' \in \ST_{1}(\gamma)$  by reversing the order of certain consecutive values in columns of $\tau$. 
Specifically, for each maximal set $\{i, i+1, \ldots, i+k-1\} \in U$, the values in $[i, i+k]$ are reversed so that $\{i+1, i, \ldots, i+k\}$ is similarly maximal in $\Des(\tau')$.
All other values are fixed. Note that since the cells of standard filling $\tau$ are identified with the values they contain, we represent $U$ by a subset of $[n]$.
Further, since $\tau\in\ST_0(\gamma)$ is the result of sorting the entries within the columns of $\tau'$, $\tau'$ has a unique preimage.

Next, for $\tau' \in \ST_{1}(\gamma)$,  let  
\begin{align*}
&\coinv(\Des(\tau')):= \sum_{u\in \Des(\tau')}\arm(u), \\
%\end{align*}{\begin{align*}
&\omega(\tau') := h(\gamma) - |\{i \in \tau' \colon i \text{ and } i+1 \text{ share a column}\}|.
\end{align*}

Lastly, we replace $V(\tau')$ with a new set in the context of $\ST_1(\gamma)$. The \emph{descent group of $i$} is the maximal connected set of cells in the column of $i$ such that every cell is a decent except the bottom cell. By construction, every cell is contained in a unique descent group. We say \emph{$i$ attacks $i+1$ through descents} if a cell in the descent group of $i$ attacks a cell in the descent group of $i+1$. Notice that if $i$ attacks $i+1$ in $\tau'\in \ST_1(\gamma)$, then $i$ must be at the top of its descent group and $i+1$ must be at the bottom of its descent group. Define $\Nu(\tau')$ as the set of $i$ in $\tau'$ such that $i\in\ID(\tau')$
or $i$ attacks $i+1$ through descents. Since $i$ attacking $i+1$ implies $i$ attacks $i+1$ by descents, it follows that $V(\tau')\subseteq  \Nu(\tau')$.  
 

\begin{example}\label{example: HLstats}
Consider $\tau \in \ST_0((1,4,3))$ and $\tau' = \iota_{\{3, 4\}}(\tau) \in \ST_1((1,4,3))$.
\begin{center}
\begin{tikzpicture}[scale=0.5]
\node at (-2,-1) {$\tau=$} ;
\cell023
\cell124\cell111
\cell225\cell212
\cell327\cell318\cell306
\end{tikzpicture} 
\qquad \qquad
\begin{tikzpicture}[scale=0.5]
\node at (-2,-1) {$\tau'=$};
\cell025
\cell124\cell111
\cell223\cell212
\cell327\cell318\cell306
\end{tikzpicture} 
\end{center}
Here, $\coinv(\Des(\tau')) = 2$, $\omega(\tau') = 2$, $\ID(\tau')=\{3,4,7\}$, and $\Nu(\tau')=\{2, 3, 4, 5, 6, 7\}$.  
\end{example}

\begin{theorem*}[\cref{thm:HL}]
Let $\gamma$ be a strong composition. Then
\begin{align*}
G_{\gamma}(X;0,t)& =
\sum_{\tau\in\ST_{1}(\gamma)}
(1-t)^{\omega(\tau)}
(-t)^{|\Des(\tau)|}
t^{\coinv(\tau)-\coinv(\Des(\tau))} 
 F_{\Nu(\tau)}.
\end{align*}
\end{theorem*}
%
%
\begin{proof}
The proof is a matter of changing the order of summations in \eqref{eq:HL}, applying $\iota_U$, tracking the changes to statistics, and combining the sums. %Reversing the order of summations gives
%
%\begin{align}\label{eq: reversed sum}
%G_{\gamma}(X;0,t) 
%= \sum_{U \subseteq \widehat{\dg}(\gamma)}
%\sum_{\tau\in\ST_0(\gamma)  \atop W(\tau)\supseteq U}  
%(1-t)^{h(\gamma)-|W(\tau)|}
%(-t)^{|U|}
%t^{\coinv(\tau)} 
%F_{V(\tau)\cup U}.
%\end{align}
%
%Next, apply $\iota_U$ to each $\tau$. The image is in $\ST_{1}(\gamma)$. Further, each tableau of  $\ST_1(\gamma)$ will appear exactly once. We now work through the changes to the factors in (\ref{eq: reversed sum}) from left to right.
%
%Consider the image of $W(\tau)$ under $\iota_U$. Because tableaux in $\ST_0(\gamma)$ have no descents, $W(\tau)$ is precisely the set containing those $i$ that share a column with $i+1$. Because $\iota_U$ permutes entries within columns, $h(\gamma) - |W(\tau)|$ is taken to $\omega(\tau)$.
%
%By the definition of $\iota_U$, $|U|$ is taken to $|\Des(\tau)|$. Thus, $(-t)^{|U|}$ is taken to $(-t)^{|\Des(\tau)|}$.
%
%Notice that every type $B$ triple with its upper right cell contained in $U$ is changed from an inversion to a coinversion. We may compensate by multiplying by 
%$t^{-\coinv(U)}$. Because the relative order of entries within all other triples is preserved, no other coinversions are changed. 
%
%Finally, $\Nu(\iota_U(\tau)) = V(\tau)\cup U$. To see this, first notice that each $i \in U$ is moved below $i+1$, and so is moved after $i+1$ in the reading word. Hence, each element of $U$ is taken to an element of $\Nu(\iota_U(\tau))$. The only other values that need to be considered are those $i$ that are in a different column than $i+1$. In this case, $i$ can only move up relative to $i+1$, and $i+1$ can only move down relative to $i$. It is thus possible that $\iota_U$ removes $i$ from $V(\tau)$, but it cannot add any values to $V(\tau)$. By construction of $\iota_U$, if $i$ and $i+1$ were attacking in $\tau$ or $i\in\ID(\tau)$, 
%then $i$ and $i+1$ are attacking through descents in $\iota_U(\tau)$. We thus replace $F_{V(\tau)\cup U}$ with $F_{\Nu(\tau)}$. Tracking these changes gives
%
%\begin{align*}
%G_{\gamma}(X;0,t) 
%= \sum_{U \subseteq \widehat{\dg}(\gamma)}
%\sum_{\tau\in\ST_{1}(\gamma)  \atop \Des(\tau) = U} 
%(1-t)^{\omega(\tau)}
%(-t)^{|\Des(\tau)|} 
%t^{\coinv(\tau)-\coinv(\Des(\tau))} 
%F_{\Nu(\tau)}.
%\end{align*}
Combining the two sums completes the proof.
\end{proof}

\subsection{Jack specialization}
We also consider the specialization of $G_{\gamma}(X;q,t)$ to the setting of Jack polynomials, from which we immediately get a new definition of a \emph{quasisymmetric Jack polynomial}. Recall that the Jack polynomial indexed by a partition $\lambda$ with parameter $\alpha$ is a symmetric polynomial that can be obtained from 
\[
J_{\lambda}(X;\alpha)=\lim_{t\rightarrow1^{-}}
\left( \prod_{u\in\dg(\lambda)} \frac{1-t^{\arm(u)+1}t^{\alpha\leg(u)}}{1-t} \right)
P_{\lambda}(X;t^{\alpha},t).
\]

\noindent
Thus define the quasisymmetric Jack polynomial indexed by a strong composition $\gamma$ as
\begin{equation}
G_{\gamma}(X;\alpha)=\lim_{t\rightarrow1^{-}} 
\left( \prod_{u\in\dg(\lambda)} \frac{1-t^{\arm(u)+1}t^{\alpha\leg(u)}}{1-t} \right)
G_{\gamma}(X;t^{\alpha},t).
\end{equation}

%\noindent
%Using \eqref{eq:G}, we get
%\begin{equation}\label{eq:jack}
%G_{\gamma}(X;\alpha)=\sum_{\sum_{\substack{T\in\NAT(\gamma)\\T\ \text{packed}}} } \left(\prod_{\substack{u \in \widehat{\dg}(\lambda)\\T(u)=T(\South(u))}} (\alpha(\leg(u)+1)+\arm(u)+1) \right)M_T. 
%\end{equation}

%\noindent
%This is a  refinement of the Knop--Sahi formula \cite[Theorem 5.1]{KS96},
%\[
%J_{\lambda}(X;\alpha)=\sum_{\substack{T\in\NAT(\lambda)}}  \left(\prod_{\substack{u \in \widehat{\dg}(\lambda)\%\T(u)=T(\South(u))}} (\alpha(\leg(u)+1)+\arm(u)+1) \right) x^T. 
%\]

\noindent
Using \cref{thm:q} we obtain the following corollary.
\begin{corollary}
The quasisymmetric Jack polynomial has the following fundamental expansion:
\begin{align*}
G_{\gamma}(X;\alpha)=&\sum_{\tau\in\ST(\gamma)}  \left( \prod_{u\in W(\tau)} (\alpha(\leg(u)+1)+\arm(u)+1) \right) \\
%&\hspace{1cm}
&\times \sum_{U \subseteq W(\tau)} (-1)^{|U|} \left( \prod_{u \in U} \frac{\alpha(\leg(u)+1)+\arm(u)}{\alpha(\leg(u)+1)+\arm(u)+1} \right)F_{V(\tau)\cup U}.
\end{align*}
\end{corollary}




\acknowledgements{The authors would like to thank Jim Haglund for bringing them together. The first and second authors thank the Institute of Mittag-Leffler for their hospitality.}

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