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%%%%% Auteur
%%%1
\author{\firstname{Per} \lastname{Alexandersson}}
\address{Dept. of Mathematics\\
Royal Institute of Technology\\
SE-100 44 Stockholm, Sweden}
\email{per.w.alexandersson@gmail.com}
%%%2
\author{\firstname{Joakim} \lastname{Uhlin}}
\address{Dept. of Mathematics\\
Royal Institute of Technology\\
SE-100 44 Stockholm, Sweden}
\email{joakim\_uhlin@hotmail.com}
%%%%% Sujet
\keywords{Cyclic sieving, Macdonald polynomials, LLT polynomials, crystals, Schur-positivity.}
\subjclass{05E10, 05E05, 06A07}
%%%%% Gestion
\DOI{10.5802/alco.123}
\datereceived{2019-08-15}
\daterevised{2020-03-25}
\dateaccepted{2020-04-12}
%%%%% Titre et résumé
\title{Cyclic sieving, skew Macdonald polynomials and Schur positivity}
\begin{abstract}
When $\lambda$ is a partition, the specialized non-symmetric Macdonald polynomial
$\mathrm{E}_{\lambda}(\mathbf{x};q;0)$ is symmetric and related to a modified Hall--Littlewood polynomial.
We show that whenever all parts of the integer partition $\lambda$ are multiples of $n$,
the underlying set of fillings exhibit the cyclic sieving
phenomenon (CSP) under an $n$-fold cyclic shift of the columns.
The corresponding CSP polynomial is given by $\mathrm{E}_{\lambda}(\mathbf{x};q;0)$.
In addition, we prove a refined cyclic sieving phenomenon where the content of the fillings is fixed.
This refinement is closely related to an earlier result by B.~Rhoades.
We also introduce a skew version of $\mathrm{E}_{\lambda}(\mathbf{x};q;0)$.
We show that these are symmetric and Schur positive via a variant of
the Robinson--Schenstedt--Knuth correspondence and we also describe crystal raising and lowering operators
for the underlying fillings.
Moreover, we show that the skew specialized non-symmetric Macdonald polynomials
are in some cases vertical-strip LLT polynomials.
As a consequence, we get a combinatorial Schur expansion of a new family of LLT polynomials.
\end{abstract}
%%%%%%%%%%%%%%%%%
\begin{document}
\maketitle
\section{Introduction}
The cyclic sieving phenomenon (CSP), introduced by V.~Reiner, D.~Stanton and D.~White~\cite{ReinerStantonWhite2004},
is currently an active research topic,
see \eg \cite{BennettMadillStokke2014,OhPark2019,Rush2018,ShenWeng2018}.
In this article, we provide families of cyclic sieving on tableaux related to
certain specializations of non-symmetric Macdonald polynomials.
This settles an earlier conjecture by the authors presented in~\cite{Uhlin2019}.
The non-symmetric Macdonald polynomials are in our case closely
related to the transformed Hall--Littlewood functions and Kostka--Foulkes polynomials,
previously studied in the CSP context by B.~Rhoades~\cite{Rhoades2010b}.
%
The family of polynomials we study is the specialization of the non-symmetric Macdonald polynomials
$\macdonaldE_\lambda(x_1,\dotsc,x_m;q,t)$ when $\lambda$ is an integer partition and $t=0$.
They can be defined as a weighted sum over certain fillings of the Young diagram $\lambda$.
We denote this set of fillings $\COF(\lambda,m)$, which is defined \cref{subsec:hallLittlewood}.
\subsection{Main results}
For an integer partition $\lambda=(\lambda_1,\dotsc,\lambda_\ell)$,
we let $n\lambda$ denote the partition $(n\lambda_1,\dotsc,n \lambda_\ell)$.
We show that there is a natural action $\phi$
on the fillings $\COF(n\lambda,m)$ where each block of $n$ consecutive columns
is cyclically rotated one step.
Consequently $\phi$ generates a $C_n$-action on $\COF(n\lambda,m)$.
%
In \cref{thm:mainCSP}, we prove that for every $n,m \in \setP$ and integer partition $\lambda$,
the triple
\begin{equation}\label{eq:cspIntro}
\left( \COF(n \lambda, m), \langle \phi \rangle , \macdonaldE_{n\lambda}(1^m; q,0) \right)
\end{equation}
exhibits the cyclic sieving phenomenon.
Moreover,
as $\lambda$ is held fixed and $n=1,2,3,\dotsc$, this family is a \emph{Lyndon-like family},
a notion by P.~Alexandersson, S.~Linusson and S.~Potka~\cite{AlexanderssonLinussonPotka2019}
(see also~\cite{Gorodetsky2019}) meaning that fixed points in $\COF(n \lambda, m)$
under $\phi^k$ are in natural bijection
with the elements in $\COF(\frac{n}{k} \lambda, m)$ whenever $k \mid n$.
%
When $\lambda = (1)$, this phenomenon reduces to a classical cyclic sieving phenomenon on
words of length $n$ in the alphabet $[m]$, see \cref{ex:CSPonwords} below.
A skew version of \eqref{eq:cspIntro} is given in \cref{thm:mainCSPSkew}.
We also prove a refined cyclic sieving phenomenon.
Let $\COF(n\lambda,\nu)$ denote the set of coinversion-free fillings with shape $n\lambda$ and content $\nu$.
In \cref{thm:refinedMacdonaldCSP}, we show that
\begin{equation}\label{eq:refinedIntro}
\left( \COF(n \lambda, \nu), \langle \phi \rangle, [\monomial_\nu]\macdonaldE_{n \lambda}(\xvec;q,0) \right)
\end{equation}
exhibits the cyclic sieving phenomenon. When $\lambda = (n)$, we recover the
cyclic sieving phenomenon on words of length $n$ with content $\nu$,
and
\[
[\monomial_\nu]\macdonaldE_{(n)}(\xvec;q,0) = \qbinom{n}{\nu}_q,
\]
a $q$-multinomial coefficient.
We remark that if we take $\lambda=(1^k)$,
$[\monomial_\nu]\macdonaldE_{n\lambda}(\xvec;q,0)$ in~\eqref{eq:refinedIntro}
can be seen as a $q$-analogue of $n$-tuples of $k$-subsets of $[m]$ with content $\nu$.
In \cref{sec:skew} we introduce a skew version of $\macdonaldE_{\lambda}(\xvec;q,0)$
and prove that these are symmetric and Schur positive.
We provide an explicit Schur expansion using a generalization of
charge in \cref{thm:skewEInSchurExpansion}.
As an application, in \cref{thm:lltNewFormula} we obtain a combinatorial
Schur expansion of a certain family of \emph{vertical-strip LLT polynomials},
which has not been considered before.
Combining \cref{thm:lltNewFormula} and \cref{thm:skewEInSchurExpansion}, we have the following main result.
\begin{theo}\label{thm:lltFromCharge}
Let $\lambda/\mu$ be a skew shape such that no column
contains more than two boxes.
Let $\nuvec$ be the tuple of skew shapes such that
$\nu_{j}$ is the vertical strip $1^{\lambda_j} /1^{\mu_j}$
and set $\alpha_i \coloneqq \lambda_i-\mu_i$.
Then
\[
\LLT_{\nuvec}(\xvec;q) =
q^{\mininv(\nuvec)} \sum_{\rho \vdash |\lambda'/\mu'|} \schurS_{\rho'}(\xvec)
\sum_{T \in \SSYT(\rho,\alpha)} q^{\charge_{\mu}(T)}
\]
where $\charge_{\mu}$ is a natural generalization of the charge statistic
defined in \cref{{def:postfixCharge}},
and $\mininv(\nuvec)$ is a simple statistic that only depends on the tuple $\nuvec$
given in \cref{eq:mininvDef}.
\end{theo}
The paper is structured as follows.
In \cref{sec:prelim}, we define the cyclic sieving phenomenon and give a brief overview of
the relevant symmetric functions.
In \cref{sec:csp1}, we give a proof of the CSP in \eqref{eq:cspIntro},
and in \cref{sec:cspRefined} we prove \eqref{eq:refinedIntro}.
%
In \cref{sec:skew} we introduce the skew specialized Macdonald polynomials
and give the Schur expansion of these.
In \cref{sec:crystalOperators}, we define crystal operators on the related
skew coinversion-free fillings and thus gives an alternative proof of
the crystal structure given in~\cite{AssafGonzalez2018}.
Finally, we prove a result in \cref{sec:llt} which implies \cref{thm:lltFromCharge}.
%
% In \cref{sec:generalizedMaj} we discuss special combinatorial
% properties of major index on $n$-tuples of $k$-subsets
% and present an intriguing connection with Touchard--Riordan polynomials.
We note that some of the results in this paper are based on earlier work done in the
second author's master's thesis~\cite{Uhlin2019}.
\section{Preliminaries}\label{sec:prelim}
\subsection{Partitions and compositions}
\begin{defi}
Let $n$ and $\ell$ be natural numbers. A \defin{weak composition} $\lambda$ of $n$
into $\ell$ \defin{parts} is defined to be an $\ell$-tuple $\lambda = (\lambda_1 ,\dotsc,\lambda_\ell)$
of non-negative integers such that $\lambda_1+\dotsb+\lambda_\ell = n$.
We say that the numbers $\lambda_1,\dotsc, \lambda_\ell$ are the \defin{parts} of $\lambda$.
If all parts of $\lambda$ are positive, we say that $\lambda$ is a \defin{composition},
and we write $\lambda \vDash n$.
If $\lambda$ has multiple parts of the same size, we may suppress them using exponents.
As an example, $(7, 7, 0, 1, 1, 1, 4, 4, 4, 4)$ may be expressed as
$(7^2, 0, 1^3, 4^4)$.
We write $m_j(\lambda)$ for the number of parts of $\lambda$
equal to $j$ and $n\lambda \coloneqq (n\lambda_1,\dotsc,n\lambda_\ell)$ for $n\in \setN$.
%
If further $\lambda_1 \geq \lambda_2 \geq \dotsb \geq \lambda_\ell$, then $\lambda$ is a \defin{partition}
of $n$, and denote this by $\lambda \vdash n$.
The \defin{parts} of $\lambda$ are the positive entries of $\lambda$.
The \defin{length} of $\lambda$ is the number of parts and is denoted $\ell(\lambda)$.
We identify partitions that only differ by trailing zeros,
so $(4,2,2,1,0,0,0)$=$(4,2,2,1,0)$=$(4,2,2,1)$ as partitions.
There is one unique partition of $0$, namely $\emptyset$ which is referred to as the \defin{empty partition}.
\end{defi}
Note that in some cases, the word \emph{parts} is ambiguous. When $\lambda$ is a weak
composition, a part can be zero whereas when $\lambda$ is a partition, a part must be a
positive integer.
This conflicting terminology is unfortunately very standard, see \eg \cite{StanleyEC2}.
\subsection{Semistandard Young tableau}
\begin{defi}
Let $\lambda=(\lambda_1, \dotsc, \lambda_\ell) \vdash n$.
The \defin{Young diagram} of $\lambda$ is defined as the set $\{(i,j) \in \setZ^2 : 1 \leq i \leq \lambda_j \}$.
%
Geometrically\footnote{Note, we use the computer-friendly matrix
indexing (row, column), which has the advantage that it is also invariant under English/French convention.}
we think of this diagram as a set of $n$ boxes with $\ell$
left-justified rows and $\lambda_i$ boxes in row $i$.
The box in position $(i, j)$ is the box in the $i^\thsup$ row
and $j^\thsup$ column.
We use the notation $\lambda$ to both refer
to the partition and to the Young diagram described by $\lambda$.
The number of boxes in a diagram $\lambda$ is denoted $|\lambda|$.
Define the \defin{conjugate} of $\lambda$, denoted $\lambda'$, to be
the Young diagram obtained by transposing $\lambda$ where the boxes
may be seen as matrix entries.
We write $\lambda' = (\lambda'_1, \dotsc, \lambda'_\ell)$.
If $\lambda$ is a partition on the form $\lambda = a^b$,
then $\lambda$ is a \defin{rectangular Young diagram}.
Throughout this article, all the diagrams are displayed
in English notation, using matrix coordinates,
with a few exceptions in \cref{sec:llt}.
\end{defi}
\begin{figure}[!ht]
\ytableausetup{boxsize=1.1em}
\[\ytableaushort{\;\;\;\;\;, \;\;\;\;\;,\;\;,\;} \qquad \ytableaushort{11123,24455, 45, 6}\]
\caption{To the left: A Young diagram of shape $\lambda=(5,5,2,1)$.
To the right: A semistandard Young tableau of shape $\lambda$.}
\label{fig: SSYT example}
\end{figure}
\begin{defi}\label{def:SSYT}
\looseness-1
Let $\lambda$ be a Young diagram.
A \defin{filling} of $\lambda$ is a map $T:\lambda \to \setP$
and a \defin{semistandard Young tableau} (SSYT) is a
filling of $\lambda$ such that in each row the entries
are weakly increasing and in each column the entries are strictly increasing.
The set of all semistandard Young tableaux of shape $\lambda$ is denoted $\SSYT(\lambda)$
and we let $\SSYT(\lambda,\mu)$ the the set of such SSYT where the number of entries equal to $i$
is given by $\mu_i$.
Let $T$ be a semistandard Young tableau. Define the \defin{reading word} of $T$,
denoted $\rw (T)$, as the word obtained by reading the entries $T$ from the
bottom row to the top row and in each row from left to right.
For example, the semistandard Young tableau
in \cref{fig: SSYT example} has reading word $\rw(T)=6452445511123$.
We let $\xvec^T \coloneqq \prod_j x_j^{m_j(T)}$ where $m_j(T)$ is the number of entries
in $T$ equal to $j$.
The semistandard Young tableau $T$ in \cref{fig: SSYT example} gives $\xvec^T=x_1^3x_2^2x_3x_4^3x_5^3x_6$.
\end{defi}
There are several equivalent ways to define the Schur functions
but the following is the most useful for our purposes.
We let the \defin{Schur function} indexed by the integer partition $\lambda$ be defined as
\[
\schurS_\lambda(\xvec) \coloneqq \sum_{T \in \SSYT(\lambda)} \xvec^T.
\]
\subsection{\texorpdfstring{$q$}{q}-analogues}
A $q$-analogue of a certain expression is a rational function in the variable $q$
from which we can obtain the original expression by letting $q \to 1$.
\begin{defi}
Let $n \in \setN$. Define the \defin{$q$-analogue of $n$} as
$[n]_q \coloneqq 1+q+\dotsb+q^{n-1}$.
Furthermore, define the \defin{$q$-factorial of $n$} as
$
[n]_q! \coloneqq [n]_q[n-1]_q \dotsm [1]_q
$.
Lastly, the \defin{$q$-binomial coefficient} is defined as
\[
\qbinom{n}{k}_q \coloneqq
\dfrac{[n]_q!}{[n-k]_q![k]_q!} \text{ if } n \geq k\geq 0, \text{ and $0$ otherwise.}
\]
\end{defi}
\begin{theo}[$q$-Lucas theorem, see \eg \cite{Sagan1992}]\label{eq:q-Lucas}
Let $n, k\in \setN$. Let $n_1, n_0, k_1, k_0$ be the unique
natural numbers satisfying $0 \leq n_0, k_0 \leq d-1$ and $n=n_1d+n_0$, $k=k_1d+k_0$. Then
\[
\qbinom{n}{k}_q \equiv \binom{n_1}{k_1}\qbinom{n_0}{k_0}_q \pmod{\Phi_d(q)}
\]
where $\Phi_d(q)$ is the $d^\thsup$ cyclotomic polynomial. In particular, we have
\begin{equation}
\qbinom{n}{k}_\xi = \binom{n_1}{k_1}\qbinom{n_0}{k_0}_\xi
\end{equation}
if $\xi$ is a primitive $d^\thsup$ root.
\end{theo}
\cref{eq:q-Lucas} will be used in later sections.
\subsection{Charge and Kostka--Foulkes polynomials}\label{sec:kostkaFoulkes}
We shall briefly describe the charge statistic and the related Kostka--Foulkes polynomials
$K_{\lambda\mu}(q)$ appearing in later sections.
This combinatorial model was first described by A.~Lascoux and M.~Schützenberger~\cite{LascouxSchutzenberger78}.
For a permutation $\sigma \in \symS_k$, let $\Des(\sigma) \coloneqq \{i \in [k-1] : \sigma_{i+1}<\sigma_{i} \}$,
the \defin{major index} be defined as $\maj(\sigma) \coloneqq \sum_{j \in \Des(\sigma)} j$,
and let $\rev(\sigma) \coloneqq (\sigma_n,\sigma_{k-1},\dotsc,\sigma_1)$ be the reverse.
%
We can now introduce the notion of \defin{charge} of a permutation.
\begin{equation}
\charge(\sigma) \coloneqq \maj(\rev(\sigma^{-1})) =
\sum_{i \notin \Des(\sigma^{-1})} (k-i).
\end{equation}
For example,
\[
\charge(198423765) = \maj(\rev(156498732) ) = \maj(237894651) = 20.
\]
We note that our way of defining $\charge$ is different from~\cite{LascouxSchutzenberger78}.
Given a word $w$ with content $\mu \vdash n$,
we partition its entries into \defin{standard subwords} as follows.
Start from the right of $w$ and mark the first occurrence of $1$.
Proceed to the left, and mark the first occurrence of $2$,
then $3$ and so on, wrapping around the end if nessecary,
until $\mu'_1$ entries have been marked.
This subword is the first standard subword of $w$.
Remove this subword, and repeat the process to find the second standard subword,
of length $\mu'_2$.
For example, the first standard subword in $w = 2 1 1 2 3 5 4 3 4 1 1 2 2 3$
has been circled.
\[
2, 1, 1, \circled{2}, 3, \circled{5}, 4, 3, \circled{4}, 1, \circled{1}, 2, 2, \circled{3}.
\]
In total, we have four standard subwords in $w$, with corresponding charge values
\[
\charge(25413) = 3,\quad
\charge(2431)=2, \quad
\charge(132)=2, \quad
\charge(12) = 1,
\]
and we define $\charge(w)$ as the sum of the charge values of the standard subwords.
In the example above, $\charge(w) = 8$.
Recall the definition of the reading word $\rw(T)$ of a semistandard Young tableau
from \cref{def:SSYT}. We then define \defin{$\charge(T) \coloneqq \charge(\rw(T))$}
and the \defin{Kostka--Foulkes polynomial} $K_{\lambda\mu}(q)$ may be computed as
\begin{equation}\label{eq:kostkaFoulkesDef}
K_{\lambda\mu}(q) = \sum_{T \in \SSYT(\lambda,\mu)} q^{\charge(T)}.
\end{equation}
\begin{example}[Computing a Kostka--Foulkes polynomial]
Consider the case $\lambda=421$, $\mu=3211$. There are four tableaux in $\SSYT(\lambda,\mu)$.
Below, these are displayed, each with the list of standard subwords
and corresponding charge values.
\begin{equation}
\substack{
\ytableaushort{1114,22,3} \\ 3214,\; 21,\; 1 \\ 1+0+0
}
\quad
%
\substack{
\ytableaushort{1113,22,4} \\ 4213,\; 21,\; 1 \\ 2+0+0
}
\quad
%
\substack{
\ytableaushort{1112,24,3} \\ 3241,\; 12,\; 1 \\ 1+1+0
}
\quad
%
\substack{
\ytableaushort{1112,23,4} \\ 4231,\; 12,\; 1 \\ 2+1+0
}
\end{equation}
Hence, $K_{\lambda\mu}(q) = q+2q^2+q^3$.
\end{example}
\subsection{Cyclic sieving}\label{ssec:csp}
\begin{defi}[Cyclic sieving, see~\cite{ReinerStantonWhite2004}]
Let $X$ be a set of combinatorial objects and $C_n = \langle g \rangle$ be the cyclic group of order $n$
acting on $X$, with $g$ as a generator.
Let $f(q)\in \setN[q]$ be a polynomial with non-negative integer coefficients.
We say that the triple $(X,C_n,f(q))$ \defin{exhibits the cyclic sieving phenomenon, (CSP)}
if for all $d \in \setZ$,
\begin{align}\label{eq:cspDef}
\#\{ x\in X : g^d \cdot x = x \} = f(\xi^d)
\end{align}
where $\xi$ is a primitive $n^\thsup$ root of unity.
\end{defi}
%
Note that it follows immediately from the definition that $\# X = f(1)$.
%
In practice, the group action of $C_n$ on $X$ and the
polynomial $f(q)$ is almost always natural in some sense.
The group action could be some form of rotation or cyclic shift of
the elements of $X$. The polynomial usually has a closed form and is
also typically the generating polynomial for some combinatorial statistic defined on $X$.
\begin{exam}[$k$-subset CSP, see~\cite{ReinerStantonWhite2004}]\label{ex:CSPonSubsets}
Let $\binom{[n]}{k}$ be the set of $k$-subsets of $[n]$.
Suppose that $C_N$ is generated by a permutation $\sigma \in \symS_n$,
where the cycles of $\sigma$ consists of $N$-cycles and one or zero singletons.
Let $C_N$ act on $[n]$ in the natural way (this is referred to as $C_N$ acting \defin{nearly freely} on $[n]$).
Then $\left(\binom{[n]}{k}, C_N, \qbinom{n}{k}_q \right)$ exhibits the cyclic sieving phenomenon.
\end{exam}
In \cref{table: instances of CSP} we summarize some of the most
famous and relevant instances of cyclic sieving.
For a more comprehensive list, see B.~Sagan's article~\cite{Sagan2011}.
\begin{table}[!ht]
\centering
\begin{tabular}{lllc}
\toprule
Set & Group action & Polynomial & Reference \\
\midrule
$k$-subsets of $[n]$ & Nearly free action & $\qbinom{n}{k}_q$ &~\cite{ReinerStantonWhite2004}\\
Words with content $\alpha$ & Cyclic shift & $\qbinom{|\alpha|}{\alpha_1, \dots, \alpha_\ell}_q$ &~\cite{ReinerStantonWhite2004}\\
Non-cross. perf. matchings & Rotation & $\frac{1}{[n+1]_q}\qbinom{2n}{n}_q$ &~\cite{ReinerStantonWhite2004}\\
$\SYT(n^m)$ & Promotion & $f^\lambda(q)$ &~\cite{Rhoades2010} \\
$01$-matrices & Shift rows/columns & See \cref{thm:rhoadesMatrices} &~\cite{Rhoades2010b}\\
\bottomrule
\end{tabular}
\caption{A\vrule height10pt depth0pt width0pt{} few known instances of cyclic sieving.
} \label{table: instances of CSP}
\end{table}
One of the main results of this paper, \cref{thm:mainCSP}, is a generalization of the first instance of cyclic sieving
in \cref{table: instances of CSP} and it is also closely related to the last instance in the table.
The situation is even more interesting when different instances of the
cyclic sieving phenomenon are related in a certain fashion.
\begin{defi}[Lyndon-like CSP,~\cite{AlexanderssonLinussonPotka2019}]\label{def:LyndonLike}
Let $\{X_n\}_{n=1}^\infty$ be a family of combinatorial objects
with a cyclic group action $C_n$ acting on $X_n$.
Furthermore, let $\{f_n(q)\}_{n=1}^\infty$ be a sequence of polynomials in $\setN[q]$,
such that for each $n=1,2,\dotsc,$ the triple $(X_n,C_n,f_n(q))$ exhibits the cyclic sieving phenomenon.
%
We say that the family of triples $\{(X_n,C_n,f_n(q))\}_{n=1}^\infty$
is \defin{Lyndon-like} if $f_{n/d}(1) = f_n(e^{\frac{2\pi i}{d}})$
for all positive integers $d$, $n$ such that $d|n$.
\end{defi}
Phrased in a different manner, the family is Lyndon-like if and only if
the number of elements in $X_n$ fixed by $g^d$ is in bijection with $X_d$
where $g$ is an element of order $n$ in $C_n$.
We note that the notion of Lyndon-like is also
studied from a different perspective (called $q$-Gauß congruences) in~\cite{Gorodetsky2019}.
\begin{exam}[{See~\cite[Prop.~4.4]{ReinerStantonWhite2004}}]\label{ex:CSPonwords}
Let $W_{nk}$ be the set of words of length $n$ in the alphabet $[k]$.
Let $C_n$ act on $W_{nk}$ by cyclic rotation.
Take $f_n(q)=\sum_{w \in W_{nk}} q^{\maj(w)}$,
where $\maj(w)$ is the sum over all indices $j$ such that $w_j>w_{j+1}$.
Then $(W_{nk}, C_n, f_n(q))$ exhibits the cyclic sieving phenomenon.
Furthermore, if we fix $k$, this family of CSP-triples is Lyndon-like.
\end{exam}
One can show that the group action on a Lyndon-like family $X_n$ corresponds to
rotation on some set of words of length $n$, see~\cite[Prop.~34]{AlexanderssonLinussonPotka2019}.
When $X_n$ is the set of binary words of length $n$,
the orbits of length $n$ are in bijection with \emph{Lyndon words}, see \oeis{A001037} in~\cite{OEIS}.
Each Lyndon-like family of combinatorial objects then has an analogue of Lyndon words.
\subsection{Burge words and RSK}\label{sec:burgeRSK}
The Robinson--Schenstedt--Knuth correspondence (RSK) is a famous combinatorial bijection with
many different applications~\cite{Krattenthaler2006,StanleyEC2}.
The version we use in this paper is a bijection between pairs of certain biwords and
pairs of semistandard Young tableaux.
We note that the biwords we consider are not lexigraphically ordered, which is otherwise typical.
\begin{defi}\label{def:burge word}
A \defin{Burge word} is a two-line array with positive integers
\[W=\begin{pmatrix}
i_1 & i_2 & \dotsb & i_m\\
j_1 & j_2 & \dotsb & j_m
\end{pmatrix} \]
sorted primarily increasingly in the first row and secondarily on the second row \emph{decreasingly}.
Furthermore, all columns are unique. As an example,
$\left(\begin{smallmatrix} 1 & 1 & 2 & 3 & 3 & 3 & 3 & 5 & 6 & 6 & 6 \\
3 & 1 & 2 & 6 & 4 & 3 & 2 & 4 & 5 & 3 & 1 \end{smallmatrix}\right)$ is a Burge word.
% In other words, the entries satisfying the conditions
% \begin{itemize}
% \item $i_1 \leq i_2 \leq \dotsb i_m$, and
% \item if $i_s=i_t$ and $sj_t$.
% \end{itemize}
A pair $(i_c, j_c)$ is called a \defin{biletter}. The first row of $W$ is called the \defin{recording word}
and the second row of the biword is called the \defin{charge word} --- the
reason for this terminology will be apparent in \cref{prop:majAspostfixCharge}.
\end{defi}
We use the same row insertion bumping algorithm as the standard
RSK on biwords, which we assume the readers are familiar with.
Our version of RSK and relevant properties is the
third variant described by C.~Krattenthaler~\cite[\S~4.3]{Krattenthaler2006}.
\ytableausetup{boxsize=0.9em}
\begin{table}[ht!]
%\renewcommand{\arraystretch}{1.7}
\begin{tabular}{lllllllll}
\toprule
\text{Inserted biletter} \vrule width 0pt height 10pt depth 8pt
& $\binom{1}{4}$ & $\binom{1}{1}$ & $\binom{2}{3}$ & $\binom{2}{2}$ & $\binom{4}{5}$ & $\binom{5}{4}$ & $\binom{5}{3}$ & $\binom{5}{1}$ \\
\midrule
$P$ & \ytableaushort{4} & \ytableaushort{1,4} & \ytableaushort{13,4} & \ytableaushort{12,3,4} & \ytableaushort{125,3,4} & \ytableaushort{124,35,4} & \ytableaushort{123,34,45} & \ytableaushort{113,24,35,4}\vrule height27pt depth22ptwidth0pt \\
\midrule
$Q$ & \ytableaushort{1} & \ytableaushort{1,1} & \ytableaushort{12,1} & \ytableaushort{12,1,2} & \ytableaushort{124,1,2} & \ytableaushort{124,15,2} & \ytableaushort{124,15,25} & \ytableaushort{124,15,25,5}
\vrule height27pt depth22pt width0pt\\
\bottomrule
\end{tabular}
\caption{Computing\vrule height 10pt width 0pt{} the image of a Burge word
under RSK via a sequence of row insertions.}
\label{table: RSK}
\end{table}
\begin{prop}\label{prop:RSK}
The RSK-algorithm yields a bijection between Burge words
and pairs of fillings $(P,Q)$ of the same shape
such that the \defin{insertion tableau} $P$
is semistandard and the \defin{recording tableau} $Q$
has the property that $Q^t$ is semistandard.
\end{prop}
As an example of \cref{prop:RSK}, the procedure in \cref{table: RSK}
shows that we have the following correspondence.
\begin{align*}
\begin{pmatrix}
1 & 1 & 2 & 2 & 4 & 5 & 5 & 5\\
4 & 1 & 3 & 2 & 5 & 4 & 3 & 1
\end{pmatrix}
\xlongrightarrow{RSK}
\left(
\ytableaushort{113,24,35,4}\; ,\quad
\ytableaushort{124,15,25,5}
\right).
\end{align*}
\subsection{Symmetric functions and plethysm}
We use standard notation (see \eg \cite{Macdonald1995,StanleyEC2})
for symmetric functions. We have the
elementary symmetric functions $\elementaryE_\lambda$,
complete homogeneous symmetric functions $\completeH_\lambda$,
the power-sum symmetric functions $\powerSum_\lambda$
and the Schur functions $\schurS_\lambda$.
Recall also the standard involution on symmetric functions~$\omega$,
with the defining properties that for $\lambda \vdash n$,
\[
\omega(\completeH_\lambda) = \elementaryE_\lambda,\qquad
\omega(\schurS_\lambda) = \schurS_{\lambda'}, \qquad
\omega(\powerSum_\lambda) = (-1)^{n-\length(\lambda) }\powerSum_\lambda.
\]
%
We shall also require a few identities
related to \defin{plethysm} --- for a comprehensive background on plethysm and
the notation used, see J.~Haglund's book~\cite{qtCatalanBook}.
In this paper, we only need the following few properties.
When $f$ is a symmetric function, we let
the \defin{plethystic substitution} $\powerSum_k[f]$ for $k \in \setN$ be defined as
\begin{align}
\powerSum_k[f] \coloneqq f(x_1^k,x_2^k,x_3^k,\dotsc).
\end{align}
Note that in particular, $\powerSum_k[\powerSum_m] = \powerSum_{km}$.
It is clear from the definition that for symmetric functions $f$ and $g$,
\[
\powerSum_k[f+g] = \powerSum_k[f] + \powerSum_k[g] \text{ and }
\powerSum_k[f\cdot g] = \powerSum_k[f] \cdot \powerSum_k[g].
\]
\begin{lemma}\label{lem:plethOmega}
For any homogeneous symmetric function $f$ of degree $n$, we have that
\[
\powerSum_k[ \omega f ] = (-1)^{(k+1)n} \omega(\powerSum_{k}[f]).
\]
\end{lemma}
\begin{proof}
Since plethysm is linear, it suffices to prove the identity for $f = \powerSum_\lambda$,
where $\lambda \vdash n$.
We have that $\powerSum_k[ \omega \powerSum_\lambda ] $ is equal to
$\powerSum_k[ (-1)^{n-\length(\lambda)} \powerSum_\lambda ]$
$= (-1)^{n-\length(\lambda)} \powerSum_{k\lambda}$
$= (-1)^{(n-\length(\lambda)) + (kn-\length(\lambda))} \omega(\powerSum_{k\lambda})$
which can be simplified to $(-1)^{(k+1)n} \omega(\powerSum_{k}[\powerSum_\lambda])$.
\end{proof}
\subsection{Hall--Littlewood and non-symmetric Macdonald polynomials}\label{subsec:hallLittlewood}
The family of non-symmetric Macdonald polynomials, $\{\macdonaldE_\alpha(\xvec;q,t)\}_\alpha$
where $\alpha \in \setN^n$ is a basis for $\setC(q,t)[x_1,\dotsc,x_n]$.
These were introduced by E.~Opdam~\cite{Macdonald1995,Opdam1995},
and further developed by I.~Cherednik~\cite{Cherednik1995nonsymmetric}.
The first definition of non-symmetric Macdonald polynomials
is quite cumbersome and indirect.
J.~Haglund, M.~Haiman and N.~Loehr~\cite{HaglundHaimanLoehr2008} found
a combinatorial formula for computing $\macdonaldE_\alpha(\xvec;q,t)$,
using the notion of \emph{non-attacking fillings},
thus generalizing F.~Knop and S.~Sahi's earlier formula for Jack polynomials~\cite{KnopSahi1997}.
In this paper, we shall only study a special case of the non-symmetric Macdonald polynomials,
namely the case when $\lambda$ is a partition and $t=0$.
Here, we use the same notation as P.~Alexandersson and M.~Sawney~\cite{AlexanderssonSawhney2017,AlexanderssonSawhney2019},
which differs slightly from Haglund et al.~\cite{HaglundHaimanLoehr2008}.
The notation $\macdonaldE_\alpha(\xvec;q,t)$ in this paper is equal to
$E_{\rev(\alpha)}(\xvec;q,t)$ in theirs where the composition has been reversed.
Since we shall only study the specialization $\macdonaldE_\lambda(\xvec;q,0)$, we do not
introduce the non-symmetric Macdonald polynomials in full generality.
Let $\lambda=(\lambda_1, \dots, \lambda_\ell)$ be a
Young diagram, $m\in \setN$ with $m\geq r$. Let $F : \lambda \to [m]$ be a filling of $\lambda$.
Three boxes $a$, $b$, $c$ in $F$ form a \defin{triple} if $a$ is just to
the left of $b$ and $c$ in a row lower than $b$ and in the same column as $b$.
The entries in a triple form an \defin{inversion-triple} if
they are ordered increasingly in a counter-clockwise orientation.
If two entries in the triple are equal, then the entry with the
largest subscript in \eqref{eq:invTriples} is considered to be the biggest.
\begin{equation}\label{eq:invTriples}
\ytableausetup{boxsize=1.1em}
\begin{ytableau}
a_3 & b_1 \\
\none[\circlearrowleft
] & \none[\scriptstyle\vdots] \\
\none & c_2 \\
\end{ytableau}
\end{equation}
%where $b$ and $c$ are in the same column, $a$ and $b$ are adjacent, and $c$ is somewhere below $c$.
%The entries in a triple form an \defin{inversion-triple} if
%\begin{equation}\label{eq:invTripCases}
%F(a) \geq F(b) > F(c), \quad F(b) > F(c) > F(a), \text{ or } F(c) > F(a) \geq F(b).
%\end{equation}
A filling of shape $\lambda$ is called a \defin{coinversion-free filling}
if every triple is an inversion-triple and the first column is
strictly decreasing from top to bottom.
The set of such fillings where the entries are in $[m]$ is denoted $\COF(\lambda,m)$,
see \cref{fig:descents} for an example.
Note that the conditions imply that every column in a coinversion-free filling
must have distinct entries.
\begin{remark}
The definition of coinversion-free filling is essentially the same as used
by P.~Alexandersson and M.~Sawhney~\cite{AlexanderssonSawhney2017} and by J.~Uhlin~\cite{Uhlin2019}
with the exception that the aforementioned texts also include \emph{basements}.
However, it is easy to see that these different definitions both yield $\macdonaldE_\lambda(\xvec;q,0)$.
Arguably, our definition makes the results in this article more natural.
S.~Assaf~\cite{Assaf2018Kostka} and S.~Assaf, N.~Gonz{\'a}les~\cite{AssafGonzalez2018}
study a generalized form of coinversion-free fillings,
which also allows composition-shaped fillings.
Therein, they are called \defin{semistandard key tabloids}.
\end{remark}
%
A \defin{descent}\footnote{Note that this seems non-standard compared to descents in words.
This terminology is due to the usage of \emph{skyline} diagrams used
when describing the non-symmetric Macdonald polynomials~\cite{HaglundHaimanLoehr2008}.
We use English notation rather than skyline diagrams.
} of a filling $F$ is a box $(i,j)$ such that $F(i,j-1)F(c_2)$ in order for the triple to be an inversion triple.
Let \defin{$\inv(F)$} denote the total number of inversion triples in $F$.
The notion of descent is extended to skew shapes, so that $(i,j)$ is a descent of $F$
if $F(i,j-1)m_{i+1}\\
F & \text{ if } m_i=m_{i+1}.
\end{cases}
\]
\end{defi}
Restricted to the set of coinversion-free fillings with $\maj=0$, the operators $\cryss_i$ are
essentially the famous Lascoux--Sch{\"{u}}tzenberger involutions~\cite{LascouxSchutzenberger78}.
The difference is that the elements with $\maj=0$ have weakly decreasing rows and strictly increasing
columns as opposed to the weakly increasing in rows
and strictly increasing columns for in semistandard Young tableaux.
It is clear by \cref{thm:majPres} that the operators $\cryss_i$ are $\maj$-preserving involutions.
Furthermore, if $F \in \COF(\lambda/\mu)$ and $\weight(F)=(w_1, \dotsc, w_\ell)$,
then $\weight(\cryss_i(F))=(w_1, \dotsc, w_{i+1},w_i, \dotsc, w_\ell)$.
This yields yet another proof that $\macdonaldE_{\lambda/\mu}(\xvec; q, 0)$ is symmetric.
In fact, it follows from general theory of crystals that
the operators $\cryss_1$, $\cryss_2$, $\dotsc,\cryss_{\ell-1}$
generate an $\symS_\ell$-action on $\COF(\lambda/\mu, \ell)$.
\section{Schur expansion of certain vertical-strip LLT polynomials}\label{sec:llt}
In this section, we briefly sketch that $\macdonaldE_{\lambda/\mu}(\xvec;q,0)$
sometimes is a \emph{vertical strip LLT polynomial}, up to a power of $q$.
As a consequence, we therefore obtain an explicit formula for the Schur expansion of
these particular LLT polynomials.
Hence, we provide a new family of LLT polynomials with a combinatorial Schur expansion,
not covered by previous results.
We note that it is a major open
problem in general to describe the LLT polynomials in the Schur basis.
\begin{defi}[As in~\cite{HaglundHaimanLoehr2005}]
Let $\nuvec$ be a $k$-tuple of skew Young diagrams. Given such a tuple, we let
$\SSYT(\nuvec) = \SSYT(\nuvec^1) \times \SSYT(\nuvec^2)\times \dotsm \times \SSYT(\nuvec^k)$
where $\SSYT(\lambda/\mu)$ is the set of skew semistandard Young tableaux of shape $\lambda/\mu$.
Given $T = (T^1,T^2,\dotsc,T^k) \in \SSYT(\nuvec)$,
let $\xvec^T \coloneqq \xvec^{T^1}\dotsm \xvec^{T^k}$ where $\xvec^{T^i}$
is the same monomial weight of $T^i$ as for Schur polynomials.
Given a cell $u = (r,c)$ (row, column) in a skew diagram, the \emph{content}
is defined as $c(u)\coloneqq c-r$.
%
Entries $T^i(u) > T^j(v)$ in a tuple form an \emph{inversion} if and only if
\[
\text{$ij$ and $c(u) = c(v)-1$}.
\]
The \defin{LLT polynomial} associated with the $k$-tuple $\nuvec$ is given by
\[
\LLT_\nuvec(\xvec;q) = \sum_{T \in \SSYT(\nuvec)} q^{\inv(T)} \xvec^T
\]
where $\inv(T)$ is the total number of inversions appearing in $T$.
One can show that $\LLT_\nuvec(\xvec;q)$ is a symmetric function, see~\cite{HaglundHaimanLoehr2005} or
\cite{AlexanderssonPanova2016} for short proofs.
\end{defi}
LLT polynomials such that each $\nuvec^j$ is a skew shape of the form $1^a/1^b$ with $a \geq b$
are called \defin{vertical strip LLT polynomials}.
%
Given a $k$-tuple $\nuvec$, we let $\mininv(\nuvec)$ be the
minimum number of inversions obtainable over all fillings.
That is,
\begin{equation}\label{eq:mininvDef}
\mininv(\nuvec) \coloneqq \min_{T \in \SSYT(\nuvec)} \inv(T).
\end{equation}
\begin{exam}
A $k$-tuple of skew shapes is traditionally illustrated using the \emph{French convention}
where box $(1,1)$ of each shape $\nuvec^i$ is placed on the line $y=x$ with content $0$,
and Cartesian coordinates are used. Below, we illustrate an element
\[
T \in \SSYT(1^3/\emptyset)\times \SSYT(1^3/1^1) \times \SSYT(1^2/1^1)\times \SSYT(1^3/\emptyset)
\]
which appears when computing the vertical-strip LLT polynomial $\LLT_\nuvec(\xvec;q)$
for $\nuvec = (111/\emptyset,111/1, 11/1, 111/\emptyset)$.
\[
T=
\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[step=1em, gray, very thin] (-0.001,0) grid (9em,9em);
\draw[gray, very thin, dashed,x=1em,y=1em] (0,0) -- (9,9);
\draw[gray, very thin, dashed,x=1em,y=1em] (0,1) -- (8,9);
\draw[gray, very thin, dashed,x=1em,y=1em] (0,2) -- (7,9);
\node[x=1em,y=1em] (9) at (6.5, 6.5) {2};
\node[x=1em,y=1em] (8) at (0.5, 0.5) {1};
\node[x=1em,y=1em] (7) at (6.5, 7.5) {4};
\node[x=1em,y=1em] (6) at (2.5, 3.5) {3};
\node[x=1em,y=1em] (5) at (0.5, 1.5) {2};
\node[x=1em,y=1em] (4) at (6.5, 8.5) {5};
\node[x=1em,y=1em] (3) at (4.5, 5.5) {5};
\node[x=1em,y=1em] (2) at (2.5, 4.5) {6};
\node[x=1em,y=1em] (1) at (0.5, 2.5) {4};
\end{tikzpicture}
\]
There are two inversions involving boxes $u$ and $v$ where $c(u)=c(v)$
and six inversions for which $c(u)=c(v)-1$.
Hence, $T$ contributes with $q^8 x_1 x_2^2 x_3x_4^2 x_5^2 x_6$.
%
The full LLT polynomial $\LLT_\nuvec(\xvec;q)$ in the Schur basis is given by
\begin{align*}
&q^8 \schurS_{333}+q^7 \schurS_{432}+(q^9+q^{10}+q^{11}) \schurS_{3222}+(q^8+2 q^9+q^{10}) \schurS_{3321} +(q^8+q^9) \schurS_{4221} \\
&+q^8 \schurS_{4311}+(q^{10}+q^{11}+q^{12}+q^{13}) \schurS_{22221} +(q^9+3 q^{10}+2 q^{11} +q^{12}) \schurS_{32211} \\
&+(q^9+q^{10}+q^{11}) \schurS_{33111} +(q^9+q^{10}) \schurS_{42111} +(2 q^{11}+2 q^{12}+q^{13}+q^{14}) \schurS_{222111} \\
&+(q^{10}+2 q^{11}+2 q^{12}+q^{13}) \schurS_{321111} +q^{11} \schurS_{411111} +(q^{12}+2 q^{13}+q^{14}+q^{15}) \schurS_{2211111} \\
&+(q^{12}+q^{13}+q^{14}) \schurS_{3111111} +(q^{14}+q^{15}+q^{16}) \schurS_{21111111}+q^{17} \schurS_{111111111}.
\end{align*}
As $q^7$ is the lowest power of $q$ that appear in the expansion, we must have that $\mininv(\nuvec)=7$.
\end{exam}
The current state-of-the-art regarding combinatorial proofs of Schur positivity
of LLT polynomials is as follows.
\begin{itemize}
\item When all shapes in $\nuvec$ are non-skew, the coefficients in the Schur basis are known to be
certain parabolic Kazhdan--Lusztig polynomials, see~\cite{LeclercThibon2000}.
Hence, the coefficients are in $\setN[q]$.
In particular, this case contains the Hall--Littlewood symmetric functions.
\item Whenever the $k$-tuple of shapes $\nuvec$ consists of at most $3$ shapes,
all avoiding an arrangement of $2\times 2$-boxes (that is, they are ribbons),
Schur positivity is given by a combinatorial formula, see J.~Blasiak~\cite{Blasiak2016}.
\item A few other cases when each shape in $\nuvec$ is a single box is given in~\cite{HuhNamYoo2020}.
\end{itemize}
\begin{theo}\label{thm:lltNewFormula}
Let $\lambda/\mu$ be a skew shape such that no column contains more than two boxes.
Then
\[
\macdonaldE_{\lambda'/\mu'}(\xvec;q,0) = q^{-\mininv(\nuvec)} \LLT_{\nuvec}(\xvec;q)
\]
where $\nuvec_{j}$ is the vertical strip $(\lambda_j) /(\mu_j)$.
\end{theo}
\begin{proof}[Proof sketch]
For the \defin{modified Macdonald polynomials} $\macdonaldH_{\lambda}(\xvec;q,t)$,
we have the symmetry $\macdonaldH_{\lambda}(\xvec;q,t) = \macdonaldH_{\lambda'}(\xvec;t,q)$.
This interchanges the r\^ole of inversion triples and major index, see~\cite{qtCatalanBook}.
This relationship extends to modified Macdonald polynomials indexed by
skew shapes $\lambda/\mu$ as long as each column contains at most two boxes,
see J.~Bandlow,~\cite[Thm.~5]{Bandlow2007}.
There is a correspondence between inversion triples and inversions
that appearing definition of LLT polynomials.
In~\cite[Eq.~(23)]{HaglundHaimanLoehr2005}, the authors
provide (via a straightforward bijective argument) an expansion of the form
\begin{align}\label{eq:macdonaldHInLLT}
\macdonaldH_{\lambda}(\xvec;q,t) = \sum_{D} q^{\maj(D)} t^{-\stat(D)} \LLT_{\nu(D)}(\xvec;t),
\end{align}
where the sum runs over all subsets (possible descents) of boxes $(i,j)$ with $i>1$ of the diagram $\lambda$.
In particular, the coefficient of the terms maximizing the major index a vertical-strip LLT polynomial.
This expansion has a natural extension to skew shapes and one can check that $\stat(\cdot)$ corresponds to $\mininv(\cdot)$
for the highest-degree term.
Combining all these observations we have
\begin{align*}
\macdonaldE_{\lambda'/\mu'}(\xvec;q,0) =
[t^\ast]\macdonaldH_{\lambda/\mu}(\xvec;q,t) =
[t^\ast]\macdonaldH_{\lambda'/\mu'}(\xvec;t,q)=
q^{-\mininv(\nuvec)}\LLT_{\nuvec}(\xvec;q).
\end{align*}
The first identity is due to \eqref{eq:macdonaldEAsSkewMacdonaldH}.
The second identity is the tricky part and relies on~\cite{Bandlow2007}.
The third identity is a simple consequence of \eqref{eq:macdonaldHInLLT}.
\end{proof}
\begin{exam}
We illustrate \cref{thm:lltNewFormula} in the case $\lambda/\mu = 4431/31$.
The skew shape $\lambda/\mu$ is illustrated in \eqref{eq:lltExample}
where we have labeled the boxes row by row,
from right to left in each row.
The corresponding $k$-tuple of vertical strips is shown to the right.
The labeling has the property that it maps \emph{inversion pairs} in the filling to the right,
to inversions in the LLT diagram, see~\cite{HaglundHaimanLoehr2005} for details.
\begin{equation}\label{eq:lltExample}
\ytableaushort{{\none}{\none}{\none}1,{\none}532,764,8}
\qquad
\qquad
\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[step=1em, gray, very thin] (-0.001,-3em) grid (11em,8em);
\draw[gray, very thin, dashed,x=1em,y=1em] (0,0) -- (8,8);
\draw[gray, very thin, dashed,x=1em,y=1em] (0,-1) -- (9,8);
\draw[gray, very thin, dashed,x=1em,y=1em] (0,-2) -- (10,8);
\draw[gray, very thin, dashed,x=1em,y=1em] (0,-3) -- (11,8);
\node[x=1em,y=1em] (1) at (0.5, 0.5) {1};
\node[x=1em,y=1em] (2) at (3.5, 3.5) {2};
\node[x=1em,y=1em] (3) at (3.5, 2.5) {3};
\node[x=1em,y=1em] (5) at (3.5, 1.5) {5};
\node[x=1em,y=1em] (4) at (7.5, 6.5) {4};
\node[x=1em,y=1em] (6) at (7.5, 5.5) {6};
\node[x=1em,y=1em] (7) at (7.5, 4.5) {7};
\node[x=1em,y=1em] (7) at (10.5, 7.5) {8};
\end{tikzpicture}
\end{equation}
Notice that no column contains more than two boxes so the conditions in the theorem applies.
The $k$-tuple $\nuvec$ is $1111/111$, $1111/1$, $111/\emptyset$, $1/\emptyset$,
and it is easy (for a computer) to check that
\begin{multline*}
\macdonaldE_{\lambda'/\mu'}(\xvec;q,0) =
\schurS_{332}+\schurS_{422}+(1+q^2) \schurS_{2222}+(2+2 q) \schurS_{3221}+\schurS_{3311}+\schurS_{4211} \\
\hspace*{20mm}+(3 q+q^2) \schurS_{22211}+4 q \schurS_{32111}+q \schurS_{41111}+4 q^2 \schurS_{221111}+3 q^2 \schurS_{311111} \\
+3 q^3 \schurS_{2111111}+q^4 \schurS_{11111111}
\end{multline*}
and that this is also equal to $q^{-1} \LLT_{\nuvec}(\xvec;q)$.
As a final check, we verify one of the coefficients with the combinatorial formula.
Using the notation in \cref{thm:skewEInSchurExpansion}, $\alpha = 1331$.
The term $(2+2q) \schurS_{3221}$ then arises from the four semistandard tableaux
\[
\substack{\ytableaushort{1222,333,4}, \\ 1} \quad
\substack{\ytableaushort{1223,233,4}, \\ 0} \quad
\substack{\ytableaushort{1223,234,3}, \\0 } \quad
\substack{\ytableaushort{1224,233,3}. \\1 }
\]
where the value of $\charge_{31}(w)=\charge(w \cdot 2111)$ is shown under each tableau.
\end{exam}
\longthanks{The authors would like to thank Svante Linusson and Samu Potka for helpful discussions.
We also thank Jim Haglund for suggesting to look at the connection with LLT polynomials
and the relevance of~\cite{Bandlow2007}. We are very grateful for the suggestions and excellent job done by the referees.
The first author was funded by the Swedish Research Council (Vetenskapsr{\r a}det), grant 2015-05308.}
\bibliographystyle{amsplain-ac}
\bibliography{ALCO_Alexandersson_349}
\end{document}