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%%%%% end tableaux macros %%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\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_1T(\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*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*