%% if you are submitting an initial manuscript then you should have submission as an option here
%% if you are submitting a revised manuscript then you should have revision as an option here
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\documentclass[finalversion]{FPSAC2023}
\articlenumber{78}
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%% note that the class file already loads {amsmath, amsthm, amssymb}

\newtheorem{thm}{Theorem}
\newtheorem{lem}{Lemma}
\newtheorem{definition}{Definition}
\newtheorem{conj}{Conjecture}
\newtheorem{example}{Example}
\newtheorem{remark}{Remark}
\newtheorem{cor}{Corollary}

%% personal packages

\usepackage{ytableau}
\usepackage{tikz,tikz-cd}

%% personal macros

\newcommand\NN{\mathbb{N}}
\newcommand\QQ{\mathbb{Q}}

\newcommand\Acal{\mathcal{A}}
\newcommand\Bcal{\mathcal{B}}
\newcommand\Hcal{\mathcal{H}}
\newcommand\Lcal{\mathcal{L}}
\newcommand\Wcal{\mathcal{W}}

\newcommand\ii{\mathfrak{I}}
\newcommand\Sym{\mathfrak{S}}

\newcommand\pHat{\widehat{\varphi}}
\newcommand\Htil{\widetilde{H}}
\newcommand\Ktil{\widetilde{K}}

\definecolor{darkred}{rgb}{0.7,0,0} % darkred color
\definecolor{darkgreen}{rgb}{0, .6, 0} % darkgreen colors

%% define your title in the usual way
\title{Foata-like bijections and science fiction}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{Sam Armon\thanks{\href{mailto:armon@usc.edu}{armon@usc.edu}.}\addressmark{1}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics, University of Southern California}

%% put the date of submission here
\received{November 15, 2022}

%% 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{A central open problem in algebraic combinatorics is to find a combinatorial formula for the Kostka-Macdonald polynomials $\widetilde{K}_{\lambda \mu}(q,t)$, which describe the expansion of the Macdonald polynomial $\widetilde{H}_\mu(Z;q,t)$ in the Schur basis. Haiman proved that $\widetilde{K}_{\lambda \mu}(q,t)$ has nonnegative integer coefficients by proving that the dimension of the Garsia-Haiman module $\Hcal_\mu$ equals $n!$, demonstrating an intricate relationship between the Kostka-Macdonald polynomials and this module.  This relationship was further expounded upon by Bergeron and Garsia, whose "science fiction" heuristics conjecture certain intersection properties of Garsia-Haiman modules which mirror observed symmetries in the Kostka-Macdonald coefficients. The most potent of these heuristics is the $\tfrac{n!}{k}$ conjecture, which asserts that the dimension of the intersection of $k$ Garsia-Haiman modules should have dimension $\tfrac{n!}{k}$. We solve the special case of the $\tfrac{n!}{k}$ conjecture where the indexing partitions have hook shape by constructing an explicit basis for the intersection, using two maps in the spirit of Foata's.}

%% put your French abstract here, or comment this out if you don't have one
% \resume{\lipsum[2]}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{Macdonald polynomials, Garsia-Haiman modules, bijective combinatorics}

%% 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; i.e., [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{macdonald.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 (transformed) Macdonald polynomials $\{ \Htil_\mu(Z;q,t) : \mu \text{ an integer partition} \}$ form a notable basis for the ring of symmetric functions, as they simultaneously generalize the Schur functions, Hall-Littlewood polynomials, and Jack symmetric functions, among others. As such, they have become a central object of study since their introduction by Macdonald in \cite{Mac88}. Of particular interest is their expansion in the basis of Schur functions $\{ s_\lambda(Z) \}$:
\[
\Htil_\mu(Z;q,t) = \sum_\lambda \Ktil_{\lambda \mu}(q,t) s_\lambda(Z).
\]
\emph{A priori} we have $\Ktil_{\lambda \mu}(q,t) \in \QQ(q,t)$, but Macdonald's \emph{positivity theorem} asserts that $\Ktil_{\lambda \mu}(q,t)$ is in fact a polynomial in $q$ and $t$ with nonnegative integer coefficients. It remains an important open problem to provide a combinatorial proof of Macdonald positivity --- e.g. writing $\Ktil_{\lambda \mu}(q,t)$ as a $(q,t)$-weighted sum over some set of combinatorial objects ---  but Haiman \cite{Hai01} resolved the positivity theorem algebraically, using a representation-theoretic analog $\Hcal_\mu$ of $\Htil_\mu(Z;q,t)$.

For $\mu$ an integer partition of $n \in \NN$, the \emph{Garsia-Haiman module} $\Hcal_\mu$ is a bigraded symmetric group module defined as follows (see \cite{GH93} for full details). Let $X = x_1, x_2, \ldots, x_n$, $Y = y_1, y_2, \ldots, y_n$, and define
\[
\ii_\mu = \{ f \in \QQ[X,Y] : f(\tfrac{\partial}{\partial x_1}, \ldots, \tfrac{\partial}{\partial x_n}; \tfrac{\partial}{\partial y_1}, \ldots, \tfrac{\partial}{\partial y_n}) \Delta_\mu = 0 \}
\]
to be the ideal of polynomials whose corresponding differential operator annihilates a certain $\Sym_n$-alternating, doubly homogeneous polynomial $\Delta_\mu \in \QQ[X,Y]$, so that $\ii_\mu$ is $\Sym_n$-invariant. The Garsia-Haiman module is defined as $\Hcal_\mu = \QQ[X,Y] / \ii_\mu$; it inherits the diagonal action of $\Sym_n$ on $\QQ[X,Y]$, and is afforded a bigrading since $\Delta_\mu$ is doubly homogeneous, so we may write
\[
\Hcal_\mu = \bigoplus_{i,j} (\Hcal_\mu)_{i,j}.
\]
Haiman made the following connection between $\Hcal_\mu$ and the Macdonald polynomials:
\begin{thm}[\cite{Hai99}]
If the dimension of $\Hcal_\mu$ is $n!$, then the bigraded Frobenius series of $\Hcal_\mu$ given by
\[
Frob_{\Hcal_\mu}(Z;q,t) = \sum_{i,j} q^it^j ch((\Hcal_\mu)_{i,j}),
\]
where $ch$ is the Frobenius map which sends the irreducible $\Sym_n$-module $S^\lambda$ to the Schur function $s_\lambda$, equals the transformed Macdonald polynomial $\Htil_\mu(Z;q,t)$.
\end{thm}
Due to known identities involving Macdonald polynomials it is necessary that $\Hcal_\mu$ afford a bigraded version of the regular representation, hence the dimension condition, but in fact this is also sufficient. 

Haiman proved that $\dim(\Hcal_\mu) = n!$ in \cite{Hai01}, thus resolving the Macdonald positivity theorem by realizing $\Ktil_{\lambda \mu}(q,t)$ as encoding the doubly graded multiplicity of the irreducible $\Sym_n$-module $S^\lambda$ in $\Hcal_\mu$. Haiman's proof, however, is algebro-geometric in nature, and does not afford an explicit basis for $\Hcal_\mu$. The construction of such a basis in general remains an open problem, although some special cases have been solved, in particular when $\mu$ has hook shape (see \cite{ARR},\cite{All},\cite{AG},\cite{Aval},\cite{GH96},\cite{Stem}). 

With an eye towards a combinatorial proof of Macdonald positivity, it is desirable to study $\Hcal_\mu$ more combinatorially. Along these lines, Bergeron and Garsia undertook a speculative study of some of the remarkable intersection properties possessed by the modules $\Hcal_\mu$ in \cite{BG}. Their most potent assertion is the following \emph{$\tfrac{n!}{k}$ conjecture}:
\begin{conj}[\cite{BG}]
Let $\lambda$ be an integer partition of $n+1$, and let $\mu^{(1)}, \ldots, \mu^{(k)}$ be partitions of $n$ each obtained from $\lambda$ by removing a removable cell from the Young diagram of $\lambda$. Then,
\[
\dim\left( \bigcap_{i=1}^k \Hcal_{\mu^{(i)}} \right) = \frac{n!}{k}.
\]
\end{conj}

In the case that $\lambda$ has hook shape --- i.e. $\lambda = (a, 1, 1, \ldots 1) = (a,1^\ell)$ for some $a \ge 2, \ell \ge 1$ --- we resolve the $\tfrac{n!}{k}$ conjecture by constructing an explicit basis for $\Hcal_{(a,1^{\ell-1})} \cap \Hcal_{(a-1,1^\ell)}$.

\section{The $\tfrac{n!}{k}$ conjecture for hook shapes}
Adin, Remmel and Roichman define a plethora of different bases for $\Hcal_\mu$ when $\mu$ has hook shape in \cite{ARR}. To tackle the $\tfrac{n!}{k}$ conjecture in this special case, we adopt a different viewpoint on their \emph{Artin basis}. This basis is indexed by permutations $w \in \Sym_n$ and each basis element encodes information about certain inversions in $w$ which depend on the partition $\mu$. We restate this basis in terms of \emph{standard fillings} of $\mu$.

Throughout, we identify a partition $\mu = (\mu_1 \ge \mu_2 \ge \cdots)$ with its Young diagram, opting for French (coordinate) notation, so that $\mu$ is depicted diagramatically as an array of bottom- and left-justified cells with $\mu_i$ cells in row $i$. We call a cell $c \in \mu$ \emph{removable} if the diagram $\mu - \{ c \}$ still has partition shape.

\begin{definition}
For any partition $\mu$ of $n \in \NN$, a \emph{standard filling} of $\mu$ is a bijective map $S : \mu \to \{ 1, 2, \ldots, n \}$, i.e. an assignment of the numbers $1, 2, \ldots, n$ to the cells of (the Young diagram of) $\mu$. The set of all standard fillings of $\mu$ is denoted $SF(\mu)$.
\end{definition}
By assigning any arbitrary order to the cells of $\mu$, we obtain a bijection between $SF(\mu)$ and $\Sym_n$, i.e. $|SF(\mu)| = n!$. For instance, the $3! = 6$ standard fillings of $\mu = (2,1)$ are:
\[
\ytableausetup{aligntableaux=bottom, boxsize=1.2em}
\begin{array}{cccccc}
\begin{ytableau}
3 \\
1 & 2
\end{ytableau},
&
\begin{ytableau}
2 \\
1 & 3
\end{ytableau},
&
\begin{ytableau}
3 \\
2 & 1
\end{ytableau},
&
\begin{ytableau}
1 \\
2 & 3
\end{ytableau},
&
\begin{ytableau}
2 \\
3 & 1
\end{ytableau},
&
\begin{ytableau}
1 \\
3 & 2
\end{ytableau}.
\end{array}
\]
For any standard filling $S$, we let $S_{i,j}$ denote the entry in row $i$, column $j$ of $S$. We now hone in on the case where $\mu = (a, 1^\ell)$ has hook shape.

\begin{definition}
Given $S \in SF(a,1^\ell)$, define a \emph{row inversion} to be a pair $S_{1,i} > S_{1,j}$ where $i < j$. Similarly, define a \emph{column inversion} to be a pair $S_{j,1} > S_{i,1}$ for $i < j$ (note the swapped indices). Denote the collection of row (resp. column) inversions in $S$ by
\begin{align*}
\mathsf{rowInv}(S) &= \{ (t,r) : t > r, t \text{ left of } r \text{ in row $1$ of } S \}, \\
\mathsf{colInv}(S) &= \{ (d,c) : d > c, d \text{ above } c \text{ in column $1$ of } S \}.
\end{align*}
\end{definition}

\begin{definition}
Given $S \in SF(a,1^\ell)$, define the polynomial $\varphi_S(X,Y) \in \QQ[X,Y]$ by
\begin{equation}\label{eq:basis-elt}
\varphi_S(X,Y) = \prod_{(d,c) \in \mathsf{colInv}(S)} x_d \prod_{(t,r) \in \mathsf{rowInv}(S)} y_r
\end{equation}
(cf. \cite{ARR}, Definition 1.6).
\end{definition}
This definition is seen to be equivalent to \cite{ARR} by associating a standard filling $S$ of $\mu$ with its reading word $w_S \in \Sym_n$. Note also that $\varphi_S(X,Y)$ encodes only the information of the \emph{larger} elements in column inversions, and of the \emph{smaller} elements in row inversions:

\begin{example}
The following standard filling of $\mu = (5,1^4)$:
\[
S =
\ytableausetup{centertableaux, boxsize = 1.2em}
\begin{ytableau}
4 \\
1 \\
9 \\
7 \\
5 & 6 & 3 & 2 & 8
\end{ytableau} 
\]
has
\begin{align*}
\mathsf{colInv}(S) &= \{ (\mathbf{\color{darkgreen} 4},1), (\mathbf{\color{darkgreen} 9},7), (\mathbf{\color{darkgreen} 9},5), (\mathbf{\color{darkgreen} 7},5) \}, \\
\mathsf{rowInv}(S) &= \{ (5,\mathbf{\color{darkred} 3}), (5,\mathbf{\color{darkred} 2}), (6,\mathbf{\color{darkred} 3}), (6,\mathbf{\color{darkred} 2}), (3,\mathbf{\color{darkred} 2}) \},
\end{align*}
so $\varphi_S(X,Y) = {\color{darkgreen} x_4x_7x_9^2}{\color{darkred} y_2^3y_3^2}$.
\end{example}

\begin{thm}[\cite{ARR}, Corollary 1.7]
The set
\[
\{ \varphi_S(X,Y) : S \in SF(a,1^\ell) \}
\]
forms a basis for the Garsia-Haiman module $\Hcal_{(a,1^\ell)}$.
\end{thm}

%\begin{remark}
%The theory of orbit harmonics developed by Garsia and Haiman (\cite{GH}) lays the groundwork for a new, direct proof that $\{ \varphi_S \}$ forms a basis for $\Hcal_\mu$, which we sketch here. Let $\alpha_1, \ldots, \alpha_n, \beta_1, \ldots, \beta_n$ be distinct rational numbers. To any standard filling $S$ we associate an \emph{orbit point} $p_S = (\alpha_{i_1}, \ldots, \alpha_{i_n}; \beta_{j_1}, \ldots, \beta_{j_n})$, and a polynomial
%\[
%\psi_S = \prod_{(d,c) \in \mathsf{colInv}(S)} (x_d - \alpha_{\mathsf{row}(c)}) \prod_{(t,r) \in \mathsf{rowInv}(S)} (y_r - \beta_{\mathsf{col}(t)}),
%\]
%so that $\varphi_S$ is the leading term of $\psi_S$. Garsia and Haiman proved that if
%\begin{enumerate}
%\item the matrix $(\psi_S(p_T))$ is nonsingular, and
%\item the Hilbert series of $\{ \psi_S \}$ is symmetric,
%\end{enumerate}
%then the leading terms $\{ \varphi_S \}$ form a basis for $\Hcal_\mu$. There exists a total ordering on standard fillings under which $(\psi_S(p_T))$ is upper-triangular with nonzero diagonal entries (hence nonsingular), and the symmetry of the Hilbert series is evidenced by the fact that $\varphi_S(q,t) = t^{n(\mu)}q^{n(\mu')} \varphi_{\widetilde{S}}(t^{-1},q^{-1})$, where $\widetilde{S}$ is obtained from $S$ by replacing the entry $i$ with $n-i+1$.
%\end{remark}
Equipped with this basis, we prove the following special case of the $\tfrac{n!}{k}$ conjecture:
\begin{thm}[$\frac{n!}{k}$ for hook shapes]\label{nfact/2}
The $\tfrac{n!}{k}$ conjecture holds for the hook shape $\lambda = (a,1^\ell)$, i.e.
\[
\dim (\Hcal_{(a,1^{\ell-1})} \cap \Hcal_{(a-1,1^\ell)}) = \frac{n!}{2}.
\]
\end{thm}
As a hook shape has at most two removable boxes (one at the end of the first row, and one at the end of the first column), this result is as strong as possible for hook shapes since the case $k \ge 3$ is trivial.

The Artin basis of \cite{ARR} is as "nice" as can be in the sense that, not only do we prove that $\Hcal_{(a,1^{\ell-1})} \cap \Hcal_{(a-1,1^\ell)}$ has the correct dimension, we actually obtain an explicit basis for the intersection by finding exactly $\tfrac{n!}{2}$ standard fillings of $(a,1^{\ell-1})$ and of $(a-1,1^\ell)$ which yield the same basis element.

To ease the notational burden, we fix once and for all the partitions $\mu = (a,1^{\ell-1})$ and $\rho = (a-1,1^\ell)$, for some $a \ge 2$ and $\ell \ge 1$. Define the following subsets of $SF(\mu)$ and $SF(\rho)$:
\begin{equation}\label{eq:subsets}
SF_<(\mu) = \{ S \in SF(\mu) : S_{1,1} < S_{1,a} \}, \quad
SF_<(\rho) = \{ T \in SF(\rho) : T_{\ell+1,1} < T_{1,1} \}.
\end{equation}
Then clearly $|SF_<(\mu)| = |SF_<(\rho)| = \tfrac{n!}{2}$. We show in turn that
\begin{equation}\label{eq:same}
\{ \varphi_S(X,Y) : S \in SF_<(\mu) \} = \{ \varphi_T(X,Y) : T \in SF_<(\rho) \}
\end{equation}
by defining a bijective map $\theta: SF_<(\mu) \to SF_<(\rho)$ which satisfies $\varphi_{\theta(S)} = \varphi_S$. 

Heuristically, the standard fillings in \eqref{eq:subsets} are the "right" candidates for a basis of $\Hcal_\mu \cap \Hcal_\rho$ because no standard filling of $\rho$ has a row inversion whose elements are $a$ units apart, and conversely no standard filling of $\mu$ has a column inversion whose elements are $\ell+1$ units apart.

As a first na\"ive step, we may define a simple auxiliary map which translates between standard fillings of $\mu$ and of $\rho$:

\begin{definition}
For any $S \in SF(\mu)$, define a standard filling $\mathsf{bump}(S) \in SF(\rho)$ by moving each entry in the first column of $S$ up one row, and then pushing each remaining entry in the first row to the left one column.
\end{definition}
Then $\mathsf{bump} : SF(\mu) \to SF(\rho)$ is a bijection whose inverse is given by first pushing the first row to the right, and then pushing the first column down. The problem, however, is that $\mathsf{bump}$ does not restrict to a bijection $SF_<(\mu) \to SF_<(\rho)$:
\begin{example}
For $\mu = (4,1^3)$ and $\rho = (3,1^4)$, we have
\[
\mathsf{bump} \left(
\ytableausetup{centertableaux, boxsize = 1.2em}
\begin{ytableau}
5 \\
1\\
2 \\
6 & 4 & 3 & 7
\end{ytableau}
\right)
=
\begin{ytableau}
5 \\
1\\
2 \\
6 \\
4 & 3 & 7
\end{ytableau}.
\]
Then $S \in SF_<(\mu)$ since $6 < 7$, yet $\mathsf{bump}(S) \not\in SF_<(\rho)$ since $5 > 4$.
\end{example}
In addition to correcting this deficiency, we must also ensure that the row and column inversions in the resulting standard filling align with the original. Thus after performing $\mathsf{bump}$, we must take the following into consideration:
\begin{enumerate}
\item There may be row inversions in $S$, created by $u = S_{1,1}$, which are no longer present in $\mathsf{bump}(S)$. We thus must rearrange the first row so as to reintroduce these lost row inversions.
\item After rearranging the first row of $\mathsf{bump}(S)$, there is a new entry $v$ in row $1$, column $1$, so we may have introduced some column inversions which were not present in $S$. Thus we must rearrange the first column both in order to negate these newly introduced column inversions, and in order to ensure that the $(\ell+1,1)$-entry is less than the $(1,1)$-entry, so that the resulting filling lies in $SF_<(\rho)$.
\end{enumerate}
To this end, we define a map $\mathsf{arm}_u$ which reintroduces the row inversions removed in 1., and a map $\mathsf{leg}_v$ which negates the column inversions introduced in 2.; by appropriately composing these maps with $\mathsf{bump}$, we obtain a map which proves \eqref{eq:same}.

\begin{definition}
For a fixed $u \in \NN$, consider the set of words
\[
\Acal_u = \{ w_1w_2\cdots w_m : m \in \NN, w_m > u \}.
\]
For any $w = w_1 w_2 \cdots w_m \in \Acal_u$, define a new word $arm_u(w)$ as follows: let $b_1, b_2, \ldots, b_j$ denote the indices for which $w_{b_i} < u$. If there are no such indices, then define $\mathsf{arm}_u(w) = w$. Otherwise,
\begin{enumerate}
\item Draw a vertical bar immediately to the left of $w_{b_i}$ if either $w_{b_i - 1} > u$, or if $i = 1$ and $b_1 = 1$. 

\item Within each newly created block which contains at least one of the $w_{b_i}$'s, move the leftmost entry which is $> u$ to the front of the block, immediately right of the (leftmost) vertical bar. 
\end{enumerate}
Define the resulting word to be $\mathsf{arm}_u(w)$.
\end{definition}
In context, for $S \in SF_<(\mu)$ and $u = S_{1,1}$, we wish to perform $\mathsf{arm}_u$ on the word $S_{1,2} S_{1,3} \cdots S_{1,a}$ consisting of the entries in the first row of $S$ which lie strictly right of the first column (note that by construction this word lies in $\Acal_u$). We then determine which entries in row $1$ of $S$ are $< u$, as these correspond precisely to the row inversions in $S$ that are lost in $\mathsf{bump}(S)$ when $u$ gets bumped into the second row. A larger number is then shuffled to the left of each of these entries in order to reintroduce the lost row inversions.

\begin{example}\label{ex:arm}
Let $w = 49263187 \in \Acal_5$. To compute $\mathsf{arm}_5(w)$ we draw vertical bars to the left of $4$, $2$, and $3$, and within each block shuffle the leftmost number $>5$ to the front:
\[
\begin{tikzpicture}
\draw[thick] (0.3,-0.3) -- (0.3,0.3);
\draw (0.3,-0.2) node (B1) {};
\draw (0.5,0) node[darkred] {$4$};
%\draw (0.5,0.5) node {${b_1}$};
\draw (1,0) node (C1) {$9$};
\draw[thick] (1.3,-0.3) -- (1.3,0.3);
\draw (1.3,-0.2) node (B2) {};
\draw (1.5,0) node[darkred] {$2$};
%\draw (1.5,0.5) node {${b_2}$};
\draw (2,0) node (C2) {$6$};
\draw[thick] (2.3,-0.3) -- (2.3,0.3);
\draw (2.3,-0.2) node (B3) {};
\draw (2.5,0) node[darkred] {$3$};
%\draw (2.5,0.5) node {${b_3}$};
\draw (3,0) node[darkred] {$1$};
%\draw (3,0.5) node {${b_4}$};
\draw (3.5,0) node (C3) {$8$};
\draw (4,0) node {$7$};
\draw [->,darkgreen] (C1.south) to [out=235,in=305] (B1.south east);
\draw [->,darkgreen] (C2.south) to [out=235,in=305] (B2.south east);
\draw [->,darkgreen] (C3.south) to [out=235,in=305] (B3.south east);

\draw (5.25,0) node {$\longmapsto$};

\draw[thick] (6.3,-0.3) -- (6.3,0.3);
\draw (6.5,0) node {$9$};
\draw (7,0) node {$4$};
\draw[thick] (7.3,-0.3) -- (7.3,0.3);
\draw (7.5,0) node {$6$};
\draw (8,0) node {$2$};
\draw[thick] (8.3,-0.3) -- (8.3,0.3);
\draw (8.5,0) node {$8$};
\draw (9,0) node {$3$};
\draw (9.5,0) node {$1$};
\draw (10,0) node {$7$};
\end{tikzpicture}
\]
So, $\mathsf{arm}_5(w) = 94628317$. Note also that in the word $5w = 549263187$, we have inversions $5 > \mathbf{\color{darkred} 4}, 5 > \mathbf{\color{darkred} 2}, 5 > \mathbf{\color{darkred} 3}, 5 > \mathbf{\color{darkred} 1}$ which are not present in $w$. However, when we perform $\mathsf{arm}_5$ on $w$, we obtain new inversions $9 > \mathbf{\color{darkred} 4}, 6 > \mathbf{\color{darkred} 2}, 8 > \mathbf{\color{darkred} 3}, 8 > \mathbf{\color{darkred} 1}$ and no others, so the smaller entries in the inversions of $5w$ and of $\mathsf{arm}_5(w)$ are the same.
\end{example}
After shuffling around the first row of $\mathsf{bump}(S)$ to correct for the lost row inversions, we must appropriately reorder the first column, for which we need the following map.

\begin{definition}
For a fixed $v \in \NN$, consider the set of words
\[
\Lcal_v = \{ w_1w_2\cdots w_m : m \in \NN, w_1 < v \}.
\]
For any $w = w_1 w_2 \cdots w_m \in \Lcal_v$, define a new word $\mathsf{leg}_v(w)$ as follows: let $c_1, c_2, \ldots, c_k$ denote the indices for which $w_{c_i} > v$. If there are no such indices, then define $\mathsf{leg}_v(w) = w$. Otherwise,
\begin{enumerate}
\item Draw a vertical bar immediately to the right of $w_{c_i}$ if either $w_{c_i + 1} < v$, or if $i = k$ and $c_k = m$. 

\item Within each newly created block which contains at least one of the $w_{c_i}$'s, move the rightmost entry which is $< v$ to the end of the block, immediately left of the (rightmost) vertical bar. 
\end{enumerate}
Define the resulting word to be $\mathsf{leg}_v(w)$.
\end{definition}

Note that $\mathsf{leg}_v$ is in a sense "dual" to $\mathsf{arm}_u$, in that $\mathsf{arm}_u$ shuffles larger entries to the left, and $\mathsf{leg}_v$ shuffles smaller entries to the right. Again, for a bit of context, we wish to perform $\mathsf{leg}_v$ on the first column of a filling after performing $\mathsf{bump}$, and then $\mathsf{arm}_{S_{1,1}}$, on some $S \in SF_<(\mu)$, when a new entry is introduced into row $1$, column $1$. Denoting the resulting filling by $T$, we will have $v = T_{1,1} < T_{2,1}$ by construction, so that $T_{2,1} T_{3,1} \cdots T_{\ell+1,1} \in \Lcal_v$, and $\mathsf{leg}_v$ then shuffles smaller numbers further up in the first column to negate the column inversions created by $v$.

\begin{example}
Let $w = 48731926 \in \Lcal_5$. To compute $leg_5(w)$ we draw vertical bars to the right of $7, 9$, and $6$, and within each block shuffle the rightmost number $< 5$ to the end:
\[
\begin{tikzpicture}
\draw (0.5,0) node (C1) {$4$};
\draw (1,0) node[darkred] {$8$};
%\draw (1,0.5) node {${c_1}$};
\draw (1.5,0) node[darkred] {$7$};
\draw[thick] (1.7,-0.3) -- (1.7,0.3);
\draw (1.7,-0.2) node (B1) {};
%\draw (1.5,0.5) node {${c_2}$};
\draw (2,0) node {$3$};
\draw (2.5,0) node (C2) {$1$};
\draw (3,0) node[darkred] {$9$};
\draw[thick] (3.2,-0.3) -- (3.2,0.3);
\draw (3.2,-0.2) node (B2) {};
%\draw (3,0.5) node {${c_3}$};
\draw (3.5,0) node (C3) {$2$};
\draw (4,0) node[darkred] {$6$};
%\draw (4,0.5) node {${c_4}$};
\draw[thick] (4.2,-0.3) -- (4.2,0.3);
\draw (4.2,-0.2) node (B3) {};
\draw [->,darkgreen] (C1.south) to [out=305,in=235] (B1.south west);
\draw [->,darkgreen] (C2.south) to [out=305,in=235] (B2.south west);
\draw [->,darkgreen] (C3.south) to [out=305,in=235] (B3.south west);

\draw (5.25,0) node {$\longmapsto$};

\draw (6 + 0.5,0) node {$8$};
\draw (6 + 1,0) node {$7$};
\draw (6 + 1.5,0) node {$4$};
\draw[thick] (6 + 1.7,-0.3) -- (6 + 1.7,0.3);
\draw (6 + 2,0) node {$3$};
\draw (6 + 2.5,0) node {$9$};
\draw (6 + 3,0) node {$1$};
\draw[thick] (6 + 3.2,-0.3) -- (6 + 3.2,0.3);
\draw (6 + 3.5,0) node {$6$};
\draw (6 + 4,0) node {$2$};
\draw[thick] (6 + 4.2,-0.3) -- (6 + 4.2,0.3);
\end{tikzpicture}
\]
So, $\mathsf{leg}_5(48731926) = 87439162$. Since in practice we read the entries in the first column from bottom to top, the \emph{coinversions} in the resulting word will correspond to column inversions. With this in mind, note that word $5w$ has coinversions $5 < \mathbf{\color{darkgreen} 8}, 5 < \mathbf{\color{darkgreen} 7}, 5 < \mathbf{\color{darkgreen} 9}, 5 < \mathbf{\color{darkgreen} 6}$ which are not present in $w$. However, when we perform $\mathsf{leg}_5$ on $w$, we  \emph{remove} the coinversions $4 < \mathbf{\color{darkgreen} 8}, 4 < \mathbf{\color{darkgreen} 7}, 1 < \mathbf{\color{darkgreen} 9}, 2 < \mathbf{\color{darkgreen} 6}$ and no others, so the larger entries in the coinversions of $5w$ and of $\mathsf{leg}_5(w)$ agree.
\end{example}

\begin{remark}
Note the similarity of $\mathsf{arm}_u$ and $\mathsf{leg}_v$ to the maps $\gamma_x$ introduced by Foata \cite{Foa}, which he used to bijectively prove that the major index and inversion number are equidistributed over $\Sym_n$. We point out that these maps are not, in fact equivalent; one immediate difference is the manner in which the respective maps factor a word $w$ into subwords. In \cref{ex:arm}, for instance, Foata's map $\gamma_5$ would factor $w$ as $\vert 49 \vert 26 \vert 3 \vert 187$ and cyclicly rotate the entries within each block. When performing $\mathsf{arm}_5$ on $w$, however, the factorization is different and the manner in which we reorder each block is as well.
\end{remark}

By composing these maps, we obtain our desired bijection $\theta : SF_<(\mu) \to SF_<(\rho)$:
\begin{definition}
Given $S \in SF_<(\mu)$, let $u = S_{1,1}$, and let $v$ be the leftmost entry in the first row of $S$ which is $> u$. Then define
\[
\theta(S) = \mathsf{leg}_v \circ \mathsf{arm}_u \circ \mathsf{bump}(S),
\]
where $\mathsf{arm}_u$ acts only on the first row of $\mathsf{bump}(S)$ --- that is, we replace the first row of $\mathsf{bump}(S)$ with $\mathsf{arm}_u(S_{1,2} S_{1,3} \cdots S_{1,a})$ --- and $\mathsf{leg}_v$ acts on the first column of $\mathsf{arm}_u(\mathsf{bump}(S))$, strictly above the first row, so that these entries are replaced by $\mathsf{leg}_v(S_{1,1} S_{2,1} \cdots S_{\ell,1})$, entered from bottom to top.
\end{definition}

Since $S_{1,a} > u$ and $u < v$ for any $S \in SF_<(\mu)$, each step in the composition is defined; thus $\theta$ is well-defined. Furthermore we have $\theta(S)_{\ell + 1,1} < v = \theta(S)_{1,1}$ by construction, so that $\theta(S) \in SF_<(\rho)$.

\begin{example}
Let $\mu = (5,1^4)$ and $\rho = (4,1^5)$, and let
\[
S =
\ytableausetup{centertableaux, boxsize = 1.2em}
\begin{ytableau}
9 \\
1 \\
7 \\
4 \\
5 & 6 & 3 & 2 & 8
\end{ytableau} \in SF_<(\mu).
\]
Then in the language of the above definition, we have $u = 5, v = 6$, so that
\begin{align*}
\theta(S) &= \mathsf{leg}_6 \circ \mathsf{arm}_5 \circ \mathsf{bump}(S) = \mathsf{leg}_6 \circ \mathsf{arm}_5\left(
\begin{ytableau}
9 \\
1 \\
7 \\
4 \\
5  \\
6 & {\color{darkred} 3} & {\color{darkred} 2} & {\color{darkgreen} 8}
\end{ytableau}
\right) \\
&= \mathsf{leg}_6\left(
\begin{ytableau}
{\color{darkred} 9} \\
{\color{darkgreen} 1} \\
{\color{darkred} 7} \\
{\color{darkgreen} 4} \\
5  \\
6 & {\color{darkgreen} 8} & 3 & 2
\end{ytableau}
\right) = 
\begin{ytableau}
{\color{darkgreen} 1} \\
9 \\
{\color{darkgreen} 4} \\
7 \\
5  \\
6 & 8 & 3 & 2
\end{ytableau} \in SF_<(\rho).
\end{align*}
Note that $\varphi_S = \varphi_{\theta(S)} = x_7^2x_9^4y_2^3y_3^2$.
\end{example}

Finally, we have:
\begin{thm}\label{l-terms}
For partitions $\mu = (a,1^{\ell - 1})$ and $\rho = (a-1, 1^\ell)$, the map $\theta : SF_<(\mu) \to SF_<(\rho)$ defined above satisfies $\varphi_S = \varphi_{\theta(S)}$ for every $S \in SF_<(\mu)$. Furthermore $\theta$ is a bijection, and $\dim (\Hcal_\mu \cap \Hcal_\rho) \ge \frac{n!}{2}$.
\end{thm}
\begin{proof}
(Sketch) By construction, we have
\[
\{ r : (t,r) \in \mathsf{rowInv}(S) \text{ for some } t \} = \{ r' : (t',r') \in \mathsf{rowInv}(\theta(S)) \text{ for some } t' \}
\]
and
\[
\{ d : (d,c) \in \mathsf{colInv}(S) \text{ for some } c \} = \{ d' : (d',c') \in \mathsf{colInv}(\theta(S)) \text{ for some } c' \},
\]
and these sets uniquely determine $\varphi_S$ (resp. $\varphi_{\theta(S)}$, so $\varphi_S = \varphi_{\theta(S)}$.

We furthermore have
\[
\theta(S) = \theta(T) \Rightarrow \varphi_{\theta(S)} = \varphi_{\theta(T)} \Rightarrow \varphi_S = \varphi_T \Rightarrow S = T,
\]
so $\theta$ is injective and therefore also bijective, thus
\[
\{ \varphi_S : S \in SF_<(\mu) \} \subset \Hcal_\mu \cap \Hcal_\rho.
\]
\end{proof}

The reverse assertion, that $\dim(\Hcal_\mu \cap \Hcal_\rho) \le \tfrac{n!}{2}$, follows from a comparatively clunkier argument.

\begin{thm}\label{at-most}
For $\mu = (a, 1^{\ell - 1}), \rho = (a-1,1^\ell)$, we have $\dim(\Hcal_\mu \cap \Hcal_\rho) \le \frac{n!}{2}$.
\end{thm}
\begin{proof}
(Sketch) We show that $\varphi_S \not\in \Hcal_\rho$ for any $S \in SF(\mu) - SF_<(\mu)$ by demonstrating that from any such polynomial $\varphi_S$, one can infer that there must be at least $a$ distinct entries in the first row of $S$, i.e. that $\varphi_S$ can not be achieved from any filling in $SF(\rho)$ (resp. $\varphi_T \not\in \Hcal_\mu$ for any $T \in SF(\rho) - SF_<(\rho)$ by an analogous argument).
\end{proof}

Combining \cref{l-terms,at-most} proves this special case of the $\tfrac{n!}{k}$ conjecture.

\begin{remark}\label{rem:other-one}
Garsia and Haiman noted in \cite{GH93} that one may define the Garsia-Haiman module as
\[
D_\mu = \text{span}\{  f(\tfrac{\partial}{\partial x_1}, \ldots, \tfrac{\partial}{\partial x_n}; \tfrac{\partial}{\partial y_1}, \ldots, \tfrac{\partial}{\partial y_n}) \Delta_\mu : f \in \QQ[X,Y] \},
\]
the span of all partial derivatives of all orders of $\Delta_\mu$, and an isomorphism $\Hcal_\mu \overset{\sim}{\to}s D_\mu$ is given by $f(X,Y) \mapsto f(\tfrac{\partial}{\partial X}, \tfrac{\partial}{\partial Y}) \Delta_\mu$.

It is worth remarking that the "science fiction" heuristics in \cite{BG}, which establish remarkable connections between Garsia-Haiman modules and the Kostka-Macdonald coefficients $\Ktil_{\lambda \mu}(q,t)$, are formulated in terms of $D_\mu$, rather than $\Hcal_\mu$. The restriction of the Artin basis to $\Hcal_\mu \cap \Hcal_\rho$ does not immediately yield a basis for $D_\mu \cap D_\rho$ in the sense that
\[
\{ \varphi_S(\tfrac{\partial}{\partial X},\tfrac{\partial}{\partial Y}) \Delta_\mu : S \in SF_<(\mu) \} \neq \{ \varphi_T(\tfrac{\partial}{\partial X},\tfrac{\partial}{\partial Y}) \Delta_\rho : T \in SF_<(\rho) \}.
\]
In fact, experimental evidence suggests that these sets are disjoint.

As the above isomorphism is bidegree-complementing, the optimistic (some would say na\"ive) conjecture would be that the complement $SF(\mu) - SF_<(\mu)$ (resp. $\rho$) indexes a basis for $D_\mu \cap D_\rho$. However, it is not true in general that $\varphi_S(\tfrac{\partial}{\partial X},\tfrac{\partial}{\partial Y}) \Delta_\mu \in D_\rho$ for arbitrary $S \in SF(\mu) - SF_<(\mu)$, so this basis is not particularly well-equipped to make this paradigm shift.
\end{remark}

\acknowledgements{I am grateful to Sami Assaf for introducing me to this problem, and for offering support along the way.}

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\end{document}
