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%% define your title in the usual way
\title{Towards Butler's conjecture}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{
Donghyun Kim\thanks{\href{mailto:hyun923010@g.skku.edu}{hyun923010@g.skku.edu}. D. Kim was supported by NRF grants partially supported by Science Research Center Program through the National Research Foundation of Korea(NRF) Grant funded by the Korean Government and NRF grants 2022R1I1A1A01070620.}\addressmark{1}, 
Seung Jin Lee\thanks{\href{mailto:lsjin@snu.ac.kr}{lsjin@snu.ac.kr}. S. J. Lee was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No. 2019R1C1C1003473).}\addressmark{2}, \and
Jaeseong Oh\thanks{\href{mailto:jsoh@kias.re.kr}{jsoh@kias.re.kr}. J. Oh was supported by KIAS Individual Grant (CG083401) at  Korea Institute for Advanced Study.
 }\addressmark{3}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics, Sungkyunkwan University, Suwon, Korea \\
\addressmark{2}Department of Mathematical Sciences, Research institute of Mathematics, Seoul National University, Seoul, Korea \\
\addressmark{3}School of Computational Sciences, Korea Institute for Advanced Study, Seoul, Korea}

%% 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{For a partition $\nu$, let $\lambda,\mu\subseteq \nu$ be two distinct partitions such that $|\nu/\lambda|=|\nu/\mu|=1$. Butler conjectured that the divided difference $\operatorname{I}_{\lambda,\mu}[X;q,t]=(T_\lambda\widetilde{H}_\mu[X;q,t]-T_\mu\widetilde{H}_\lambda[X;q,t])/(T_\lambda-T_\mu)$ of modified Macdonald polynomials of two partitions $\lambda$ and $\mu$ is Schur positive. By introducing a new LLT equivalence called column exchange rule, we give a combinatorial formula for $\operatorname{I}_{\lambda,\mu}[X;q,t]$, which is a positive monomial expansion. We also prove Butler's conjecture for some special cases.
}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{Butler's conjecture, Science Fiction  conjecture, Modified Macdonald polynomials, Macdonald intersection polynomials, column exchange rule, Butler permutations, $(q,t)$-Kostka polynomial}

%% 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
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\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}

In his seminal paper \cite{Mac88}, Macdonald introduced the \emph{Macdonald $P$-} \emph{polynomials} $P_\mu[X;q,t]$ which are $(q,t)$-extension of Schur functions indexed by partitions $\mu$. The \emph{modified Macdonald polynomials} $\widetilde{H}_\mu[X;q,t]$ are introduced as a combinatorial version of the Macdonald $P$-polynomials. 

For a partition $\mu\vdash n$, the Garsia--Haiman module $V_\mu$ is defined by the subspace in the polynomial ring $\mathbb{C}[x_1,\dots,x_n,y_1,\dots,y_n]$
spanned by partial derivatives of the polynomial $\Delta_\mu$ analogous to the Vandermonde determinant:
\[
    \Delta_\mu:=\det\left[x_i^{p_j}y_i^{q_j}\right]_{i,j=1,\dots,n},
\]
where $(p_j,q_j)$ runs over cells in $\mu$ and the symmetric group $\mathfrak{S}_n$ acts diagonally permuting $x$ and $y$ variables \cite{GH93}. Later, Haiman \cite{Hai01} proved that the Garsia--Haiman module plays a role as a representation-theoretic model for the modified Macdonald polynomial, i.e., the (bigraded) Frobenius characteristic of $V_\mu$ coincides with $\widetilde{H}_\mu[X;q,t]$.


Bergeron and Garsia \cite{BG99} studied relationships among Garsia--Haiman modules called \emph{Science Fiction conjecture}. Among the variety of its implications, they studied the intersection of Garsia--Haiman modules. To elaborate, Science Fiction conjecture implies the following. 
\begin{conj} Let $\nu$ be a partition and $\mu^{(1)},\dots,\mu^{(k)}\subseteq \nu$ be $k$ distinct partitions such that $|\nu/\mu^{(1)}|=\cdots=|\nu/\mu^{(k)}|=1.$
Then the bigraded $\mathfrak{S}_n$-module $\bigcap_{i=1}^{k} V_{\mu^{(i)}}$ is of dimension $\frac{n!}{k}$, and its Frobenius characteristic is 
\begin{equation}\label{eq: Frob = SF}
    \Frob\left(\bigcap_{i=1}^{k} V_{\mu^{(i)}},q,t\right) = \sum_{i=1}^{k} \prod_{j\neq i}\dfrac{T_{\mu^{(j)}}}{(T_{\mu^{(j)}}-T_{\mu^{(i)}})}\widetilde{H}_{\mu^{(i)}}[X;q,t],
\end{equation}
where $T_\mu:=\prod_{(i,j)\in\mu}t^{i-1}q^{j-1}$.
\end{conj} 
The first assertion is called  \emph{$\frac{n!}{k}$-conjecture}, and Armon recently proved the $\frac{n!}{2}$-conjecture for hook shapes \cite{Arm22}. The second implication of Science Fiction conjecture gives a formula for the Frobenius characteristic of the intersection of the Garsia--Haiman modules as a linear combination of the modified Macdonald polynomials. We will call the symmetric function in the right-hand side of \eqref{eq: Frob = SF} an \emph{Macdonald Intersection polynomial} of $\mu^{(1)},\dots,\mu^{(k)}$ and denote it by $\I_{\mu^{(1)},\dots,\mu^{(k)}}[X;q,t]$. In the companion paper of the authors \cite{KLO23+}, we study a remarkable connection between $\I_{\mu^{(1)},\dots,\mu^{(k)}}[X;q,t]$, $\nabla e_{k-1}$ which is the Frobenius characteristic of the diagonal harmonics \cite{Hai02}, and the Shuffle formula $D_{k-1}[X;q,t]$ in \cite{HHLRU05, CM18}. In this paper, we focus more on the case when $k=2$, which is related to Butler's conjecture. In 1994, Butler observed a surprising behavior of the modified Macdonald polynomials.
\begin{conj}(Butler's conjecture \cite{But94})
Let $\nu$ be a partition and $\lambda,\mu\subseteq \nu$ be two distinct partitions such that $|\nu/\lambda|=|\nu/\mu|=1$. Then the Macdonald intersection polynomial $\I_{\lambda,\mu}[X;q,t]=\frac{T_\lambda\widetilde{H}_\mu[X;q,t]-T_\mu\widetilde{H}_\lambda[X;q,t]}{T_\lambda-T_\mu}$ is Schur positive.
\end{conj}
Note that Science Fiction conjecture implies Butler's conjecture, as mentioned in \cite{BG99}. To be more precise, the Macdonald intersection polynomial $\I_{\lambda,\mu}[X;q,t]$ is given as a Frobenius characteristic of an $\mathfrak{S}_n$--module $V_\lambda \cap V_\mu$, thus Schur positive. 

We divide the main results in this paper into three parts. The first part gives a combinatorial formula ($F$-expansion, monomial expansion) for $\I_{\lambda,\mu}[X;q,t]$ (\cref{thm: first main F-expansion}, \cref{Cor: m-expansion}). The second part is to prove partial cases for Butler's conjecture (\cref{thm: second main s-positivities}, \cref{Cor: Schur positivity at t=1}). The third part provides combinatorial formulas for $(q,t)$-Kostka polynomials which are consistent with Butler's conjecture (\cref{thm: combinatorial formula for two column,thm: formula for hook}).



\section{Generalization of modified Macdonald polynomials}\label{Sec: generalization of Macdonald}

A (general) \emph{diagram} $D$ is a collection of points (cells) in $\mathbb{Z}_+ \times \mathbb{Z}_+$, and a \emph{bottom cell} of $D$ is a cell located in the lowest position on each column. A \emph{filled diagram} $(D,f)$ is a diagram together with a filling $$f:D\setminus\{\text{bottom cells of $D$}\}\rightarrow \mathbb{F}$$
that assigns a scalar in $\mathbb{F}$ for each cell of $D$ that is not a bottom cell. Throughout this paper, we let $\mathbb{F}=\mathbb{C}(q,t)$.  We visualize $(D,f)$ by writing the corresponding value of $f$ for each cell. For example, see \cref{Fig: filled diagram}.

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\end{figure}

We denote a diagram by its row numbers for each column: $D=[D^{(1)},D^{(2)},\dots,]$, where $D^{(j)}$ is the set of $i's$ such that $(i,j)\in D$. Throughout this paper, we assume that for every diagram $D$, each column $D^{(i)}$ is an interval $[a,b]=\{a,a+1,\dots,b\}$ for some $a\leq b$. For example, the diagram in \cref{Fig: filled diagram} can be represented as $D=[[1,3],[2,3],[2,2]]$, and the Young diagram for a partition $\mu=(\mu_1,\dots,\mu_\ell)$ can be written as $\mu=[[\mu'_1],\dots,[\mu'_{\mu_1}]]$, where $\mu'$ is the conjugate of $\mu$. 

\begin{definition} \label{def: stat definition}We give a total order on cells in $D$ row by row,
top to bottom, and left to right within each row. We denote such total order with $N_D: D \rightarrow [|D|]$. For a filled diagram $(D,f)$, we define functions $\inv_{D}:\mathfrak{S}_{|D|}\rightarrow \mathbb{F}$ and $\maj_{(D,f)}:\mathfrak{S}_{|D|}\rightarrow \mathbb{F}$ as follows: For a permutation $w \in \mathfrak{S}_{|D|}$, we say that a pair $(u,v)$ of cells in $D$ is an \emph{inversion} with respect to $w$ if $w_{N_D(u)}>w_{N_D(v)}$ and either
\begin{itemize}
    \item $u=(i,j)$, $v=(i,j')$ and $j<j'$ or
    \item $u=(i,j)$, $v=(i-1,j')$ and $j>j'$. 
\end{itemize}
Then we define 
\begin{equation*}
\inv_{D}(w):=\prod_{(u,v)}q,
\end{equation*}
where the product is over all pairs $(u,v)$ of cells in $D$ that are inversions with respect to $w$. For a permutation $w \in \mathfrak{S}_{|D|}$, we say that a cell $u=(i,j)$ in $D$ is a \emph{descent} with respect to $w$ if $w_{N_D(u)}>w_{N_D(v)}$ where $v=(i-1,j)$ is the cell just below $u$. Then we define 
\begin{equation*}
\maj_{(D,f)}(w):=\prod_{u} f(u)
\end{equation*}
where the product is over all cells that are descents with respect to $w$. Finally we define a function $\stat_{(D,f)}: \mathfrak{S}_{|D|}\rightarrow \mathbb{F}$ as 
\begin{equation*}
\stat_{(D,f)}(w):=\inv_{D}(w)\maj_{(D,f)}(w).
\end{equation*}
We defined $\stat_{(D,f)}(w)$ for a permutation $w\in\mathfrak{S}_{|D|}$. However, we note that $\stat_{(D,f)}(w)$ can be defined in the same way for any length $|D|$ word of positive integers $w$. We define a generalization of the modified Macdonald polynomial $\widetilde{H}_{(D,f)}$ for a filled diagram $(D,f)$ as 
\[
    \widetilde{H}_{(D,f)}[X;q,t] = \sum_{w\in \mathfrak{S}_n} \stat_{(D,f)}(w) F_{\iDes(w)},
\]
where $F_{S}[X]$ is the fundamental quasisymmetric functions.
\end{definition}

Given a partition $\mu$, we define the \emph{standard filling} of $\mu$, \[f^{\st}_{\mu}: \mu\setminus\{\text{bottom cells of $\mu$}\} \rightarrow \mathbb{F}\] by
\(f^{\st}_{\mu}(u)=q^{-\arm_\mu(u)}t^{\leg_\mu(u)+1}.
\) Then the modified Macdonald polynomial for the filled diagram $(\mu,f^{\st}_\mu)$ is the usual modified Macdonald polynomial
\[
\widetilde{H}_{(\mu,f^{\st}_\mu)}[X;q,t] = \widetilde{H}_\mu[X;q,t],
\]
by Haglund--Haiman--Loehr formula \cite{HHL05}.

\begin{example}
Let $(D,f)$ be a filled diagram as depicted below,
\[
    \ytableausetup{boxsize=1.6em, boxframe=0.1em}
    \begin{ytableau}
    \textcolor{white}{0} & \alpha & \none\\
    \none &  & \none
    \end{ytableau},
\]
where $\alpha\in\mathbb{F}$. Then the statistics are given in the following table.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
    \hline
    $w \in \mathfrak{S}_3$ & 123 & 132 & 213 & 231 & 312 & 321\\
    \hline
    $\iDes(w)$ & $\emptyset$ & $\{2\}$ & $\{1\}$ & $\{1\}$ & $\{2\}$ & $\{1,2\}$ \\
    \hline
    $\inv_D(w)$ & 1 & 1 & $q$ & 1 & $q$ & $q$\\
    \hline
    $\maj_{(D,f)}(w)$ & 1 & $\alpha$ & 1 & $\alpha$ & 1 & $\alpha$\\
    \hline
\end{tabular}
\end{center}
Thus, the modified Macdonald polynomial for $(D,f)$ is
\[
    \widetilde{H}_{(D,f)}[X;q,t] = F_{\emptyset} + (q+\alpha)F_{\{1\}} + (q+\alpha)F_{\{2\}} + q\alpha F_{\{1,2\}}.
\]


Two distinct filled diagrams $(D,f)$ and $(D',f')$ can give the same modified Macdonald polynomials. For example, given a filled diagram $(D,f)$ where 
$$
    D=[D^{(1)},D^{(2)},\dots,D^{(\ell)}].
$$
Consider a diagram
$$
    D'=[D^{(2)},\dots,D^{(\ell)},D^{(1)}+1],
$$
where $I+1=\{a+1:a\in I\}$ for an interval $I$. In other words, $D'$ is a diagram obtained by moving the leftmost column of $D$ to the end of the right and placing it one cell up. Cells in $D$ and $D'$ are naturally in bijection, and we give a filling $f'$ on $D'$ inherited from the filling $f$ on $D$. We define $\cycling (D,f):=(D',f')$. Then the following is well-known. 
\begin{lem}\label{lem: cycling}(Cycling rule)
Let $(D,f)$ be a filled diagram and $(D',f')=\cycling(D,f)$. Then we have
\[
    \widetilde{H}_{(D,f)}[X;q,t] = \widetilde{H}_{(D',f')}[X;q,t].
\]
\end{lem}

In \cref{Sec: Butler permutations main Lemmas} and in the omitted proofs in \cref{Sec: Proof of Schur positivities}, we exhibit a variety of relations between the modified Macdonald polynomials  (or LLT polynomials).
\end{example}



\section{Column exchange rule and Butler permutations}\label{Sec: Butler permutations main Lemmas}

\subsection{Column exchange rule}
We introduce the \emph{column exchange rule}, which concerns the equidistribution of two statistics over permutations. We use this rule to exchange the columns of filled diagrams.
\begin{proposition}\label{lem: column exchange}(column exchange rule)
Let $(D,f)$ and $(D',f')$ be filled diagrams that are the same except for the $i$-th and $i+1$-th columns. In addition, let $i$-th, and $i+1$-th column of $(D,f)$ is of the form $\mu=[[n],[m]]$ and of $(D',f')$ is of the form $\lambda = [[m], [n]]$ for some $n>m$, and $f_\mu:\mu\rightarrow\mathbb{F}$ and $f_\lambda:\lambda\rightarrow\mathbb{F}$ be fillings such that 
\begin{itemize}
    \item $f_\mu(i,1) = f_\lambda(i,2)$ for $i>m+1$,
    \item $\alpha := q^{-1}f_\mu(m+1,1) = f_\lambda(m+1,2)$, and
    \item $f_\mu(i,1) = \alpha f_\mu(i,2) = \alpha f_\lambda(i,1) = f_\lambda(i,2)$ for $ 1<i\le m$,
\end{itemize}
Then there is a $\stat$, $\iDes$, and content-preserving bijection $\phi_i^{(D,f)}$ from $\mathfrak{S}_{|D|}$ to $\mathfrak{S}_{|D'|}$. In particular, we have
\[ 
\widetilde{H}_{(D,f)}[X;q,t]= \widetilde{H}_{(D',f')}[X;q,t].
\]
\end{proposition}
For example, filled diagrams $(\mu,f_\mu)$ and $(\lambda,f_\lambda)$ depicted in \cref{fig: column exchange rule} satisfy the conditions in \cref{lem: column exchange}. In what follows, we abuse our notation to write $\phi_i^{(D,f)}=\phi_i$ for convenience.
\begin{figure}
    \centering
    \[
        \ytableausetup{boxsize=2.6em, boxframe=0.1em}
        (\mu^{(n)},f_{\mu^{(n)}}) = \begin{ytableau}
        q\alpha  \\
        \alpha a_1 & a_1 \\
        \alpha a_2 & a_2 \\
        \ldots & \ldots \\
        \alpha  a_{n-2} & a_{n-2} \\
         & 
        \end{ytableau}
        \qquad \qquad (\lambda^{(n)}, f_{\lambda^{(n)}}) = \begin{ytableau}
        \none & \alpha  \\
        a_1 & \alpha a_1 \\
        a_2 & \alpha a_2 \\
        \ldots & \ldots \\
         a_{n-2} & \alpha a_{n-2} \\
         & 
        \end{ytableau}
    \]
    \caption{Filled diagrams $(\mu,f_\mu)$ and $(\lambda, f_{\lambda})$}
    \label{fig: column exchange rule}
\end{figure}

\begin{rmk} When $(D,f)=(\mu,f^{\st}_\mu)$ for a partition $\mu$ and standard filling $f^{\st}_\mu$, the implication
\[
    \widetilde{H}_{(D,f)}[X;q,t] = \widetilde{H}_{(D',f')}[X;q,t]
\]
of the above Proposition follows from \cite[Theorem 5.1.1]{HHL08}. Our proof gives a bijective proof of their theorem, answering a question of Haglund \cite{AS17}. In addition, we would like to highlight a few differences and advantages of the column exchange rule (and its proof). 

First of all, the bijection $\phi$ preserves not only $\stat$, $\iDes$, but also content. This fact will be a crucial point in \cref{Sec: Proof of Schur positivities}, and in the upcoming paper \cite{KLO23+}. Moreover, the bijections $\phi$ and $\psi$ appear in the construction of the map $\zeta$, which will be the main ingredient of \cref{lem: main LLT lemma}. Finally, the conditions in \cref{lem: column exchange} is more flexible than the conditions in \cite[Theorem 5.1.1]{HHL08}. 
\end{rmk}

\subsection{Butler permutations}
For a directed perfect matching $M=\{(a_1,a_2), \dots, (a_{2n-1},a_{2n})\}$, we say an arc $(a,b)$ is in the \emph{forward direction} if $a<b$ and in the \emph{reverse direction} otherwise. An arc $(a,b)$ is \emph{nested} by the other arc $(c,d)$ if $\min\{c,d\}<a,b$ and $\max\{c,d\}>a,b$. We say that the pair of arcs $(a,b)$ and $(c,d) $ is \emph{crossing} if $\min\{a,b\}<\min\{c,d\}<\max\{a,b\}<\max\{c,d\}$ or $\min\{c,d\}<\min\{a,b\}<\max\{c,d\}<\max\{a,b\}$. If not, we say the pair is \emph{noncrossing}.
\begin{definition}
For a permutation $w=w_1\dots w_{2n} \in \mathfrak{S}_{2n}$. Consider a directed perfect matching $M(w)$ on $\{0,1,\dots,2n+1\}$ with directed arcs $\alpha_1(w),\alpha_2(w),\dots$ as follows:
\begin{itemize}
    \item $\alpha_1(w)$ is a directed arc from $w_{n}$ to $n+1$,
    \item $\alpha_i(w)$ is a directed arc from $w_{n-2(i-1)}$ to $w_{n-2(i-1)+1}$ for $2\leq i\leq \frac{n}{2}$, and
    \item $\alpha_{\frac{n}{2}+1}(w)$ is a directed arc from 0 to $w_1$.
\end{itemize}
Let $k(w)$ the smallest integer $k$ such that the pair of arcs $\alpha_k(w)$ and $\alpha_{k+1}(w)$ is noncrossing if exists. We say that $w$ is a \emph{Butler permutation} if either
\begin{itemize}
    \item $\alpha_{k+1}(w)$ is nested by $\alpha_{k}(w)$ and $\alpha_k(w)$ is in the reverse direction, or
    \item $\alpha_{k+1}(w)$ is not nested by $\alpha_{k}(w)$ and $\alpha_k(w)$ is in the forward direction.
\end{itemize}
If there is no such $k$, we let $k(w)$ to be $\frac{n}{2}$. We say that $w$ is a \emph{Butler permutation} if $\alpha_{k(w)}$ is in the reverse direction. We denote the set of Butler permutations of length $2n$ by $\mathfrak{B}_{2n}$.
\end{definition}

Let $w$ be a word of weight $\alpha=(\alpha_1,\alpha_2,\dots)$, i.e. number of $i$'s in $w$ is $\alpha_i$. The \emph{standardization} $\std(w)$ of $w$ is given by replacing $1$'s with the numbers $[1,\alpha_1]$ in increasing order from left to right, replacing $2$'s with the numbers in $[\alpha_1+1,\alpha_1+\alpha_2]$ in increasing order from left to right, and so on. For example, we have $\std(3155)=2134$.

\subsection{Main Lemma}
\begin{proposition}\label{lem: main LLT lemma}
Suppose $n>m$. Let $\bar{\mu}=[[n],[m]\setminus\{1\}]$, $\bar{\nu}=[[m+1],[m]\setminus\{1\}]\subseteq\mu$, and $\bar{\lambda} = [[m], [n]\setminus\{1\}]$. Let $f_{\bar{\mu}}:\bar{\mu}\rightarrow\mathbb{F}$ and $f_{\bar{\lambda}}:\bar{\lambda}\rightarrow\mathbb{F}$ be fillings such that 
\begin{itemize}
    \item $f_{\bar{\mu}}(i,1) = f_{\bar{\lambda}}(i,2)$ for $i>m+1$,
    \item $\alpha := q^{-1}f_{\bar{\mu}}(m+1,1) = f_{\bar{\lambda}}(m+1,2)$,
    \item $f_{\bar{\mu}}(i,1) = \alpha f_{\bar{\mu}}(i,2) = \alpha f_{\bar{\lambda}}(i,1) = f_{\bar{\lambda}}(i,2)$ for $ 2<i\le m$, and
    \item $f_{\bar{\mu}}(2,1)=\alpha f_{\bar{\lambda}}(2,1)$.
\end{itemize}
Then, for any filled diagram $(D,f_D)$,
\[ 
    \dfrac{\widetilde{H}_{((D,\bar{\mu}),(f_D,f_{\bar{\mu}}))}[X;q,t] - \alpha \widetilde{H}_{((D,\bar{\lambda}),(f_D,f_{\bar{\lambda}}))}[X;q,t]}{1-\alpha} = \sum_{w} \stat_{(\bar{\mu}, f_{\bar{\mu}})}F_{\iDes(w)},    
\]
where the sum is over all permutations $w\in\mathfrak{S}_{n+m-1+|D|}$ such that $\std(w{\downarrow}^{(D,\bar{\mu})}_{\bar{\mu}})\in \mathfrak{B}_{2m}$ .
\end{proposition}




\section{Combinatorial formulas for $\I_{\lambda,\mu}[X;q,t]$.}\label{Sec: proof of main theorem}

\subsection{Proof of a combinatorial formula for the $F$-expansion for $\I_{\lambda,\mu}[X;q,t]$.}
\begin{thm}\label{thm: first main F-expansion} For a partition $\nu$, let $\mu,\lambda\subseteq \nu$ be two distinct partitions such that $|\nu/\mu|=|\nu/\lambda|=1$. Moreover, suppose the cells $\nu/\mu$, and $\nu/\lambda$ are in $i$-th and $j$-th column respectively for $i<j$. Then the fundamental quasisymmetric expansion of $\I_{\lambda,\mu}[X;q,t]$ is given by
\begin{equation}
    \I_{\lambda,\mu}[X;q,t] = \sum_{w \in \mathcal{B}_{\mu,\lambda}} \stat_{(\mu,f^{\st}_\mu)}(w) F_{\iDes(w)}[X].
\end{equation}
Here, $\mathcal{B}_{\mu,\lambda}$ is the set of permutations $w$ of $\mathfrak{S}_{|D|}$ such that restriction of 
\[
    \std( \phi_{\nu_1-1}\cdots \circ \phi_j \circ \phi_1\circ \cdots \circ \phi_{i-1}(w))
\]
to the first $m+1$ rows of the last two columns is a Butler permutation.
\end{thm}

We sketch a proof of \cref{thm: first main F-expansion} by giving an example. Let $\lambda = [[4],[3],[3],[2],[1]]$ and $\mu=[[4],[4],[3],[1],[1]]$. First of all, the filled diagram $(\lambda,f^{st}_\lambda)$ is 
\[
    \ytableausetup{boxsize=2.8em, boxframe=0.1em}
    \begin{ytableau}
    t  \\
    q^{-2}t^2 & q^{-1}t & t \\
    q^{-3}t^3 & q^{-2}t^2 & q^{-1}t^2 & t\\
     & & & &
    \end{ytableau}.
\]
After applying the column exchange rule to move the second column to the left and the fourth column to the right, then cycling rule to move the first column to the last, we obtain the filled diagram $(D_\lambda,f_{D_\lambda})$
\[
    \begin{ytableau}
    q^{-1}t  & \none & \none & \none & q^{-1}t\\
    q^{-2}t^2 & t & \none & \none & q^{-2}t^2\\
    q^{-3}t^3 & q^{-1}t^2 & \none & q^{-1}t & \\
     & & & 
    \end{ytableau}.
\]
On the other hand, the filled diagram $(\mu,f^{st}_\mu)$ is 
\[
    \ytableausetup{boxsize=2.8em, boxframe=0.1em}
    \begin{ytableau}
    q^{-1}t & t  \\
    q^{-2}t^2 & q^{-1}t^2 & t \\
    q^{-2}t^3 & q^{-1}t^3 & t^2\\
     & & & & 
    \end{ytableau}.
\]
After applying the column exchange rule to move the fourth column to the left and the second column to the right, then cycling rule to move the first column to the last, we obtain the filled diagram $(D_\mu,f_{D_\mu})$
\[
    \begin{ytableau}
    q^{-1}t  & \none & \none & q^{-1}t & \none \\
    q^{-2}t^2 & t & \none & q^{-1}t^2\\
    q^{-3}t^3 & q^{-1}t^2 & \none & q^{-3}t^3 & \\
     & & & 
    \end{ytableau}.
\]
One can check that the filled diagrams $(D_\lambda,f_{D_\lambda})$ and $(D_\mu,f_{D_\mu})$ are the same except for the last two columns. In addition, the last two columns satisfy the condition of \cref{lem: main LLT lemma} where $\alpha=q^2t^2$.

We end this section with two implications of \cref{thm: first main F-expansion}. The first one is a monomial expansion of $\I_{\lambda,\mu}[X;q,t]$, and the second one is about the specialization $\I_{\lambda,\mu}[X;1,1]$.

\begin{corollary}\label{Cor: m-expansion}
For a partition $\nu$, let $\mu,\lambda\subseteq \nu$ be two distinct partitions such that $|\nu/\lambda|=|\nu/\mu|=1$. Then the monomial expansion of $\I_{\lambda,\mu}[X;q,t]$ is given by
\begin{equation}
    \I_{\lambda,\mu}[X;q,t] = \sum_{\nu\vdash n}\sum_{w \in \mathcal{B}_{\mu,\lambda,\nu}} \stat_{(\mu,f^{\st}_\mu)}(w)m_\nu[X].
\end{equation}
Here, $\mathcal{B}_{\mu,\lambda,\nu}$ is the set of words $w$ of weight $\nu$ whose standardization $\std(w)$ is a Butler permutation of $(\mu,\lambda)$.
\end{corollary}

\begin{corollary}\label{Cor: h21111 at q=t=1}
For a partition $\nu$, let $\mu,\lambda\subseteq \nu$ be two distinct partitions such that $|\nu/\lambda|=|\nu/\mu|=1$. Then we have 
\[
    \I_{\lambda,\mu}[X;1,1]=h_{(2,1^{n-2})}[X],
\]
which is independent of $\mu$ and $\lambda$. In particular, this is consistent with $\frac{n!}{2}$-conjecture.
\end{corollary}




\section{Schur positivity of $\I_{\lambda,\mu}[X;q,t]$}\label{Sec: Proof of Schur positivities}

The first Schur positivity result is the Schur positivity of $\I_{\lambda,\mu}[X;q,t]$ when we move a cell in the first or the second row of $\lambda$ to obtain $\mu$. The proof of the following theorem relies on the theory of LLT polynomials \cite{LLT97}. More precisely, we used Schur positivity of LLT polynomials \cite{GH07} and LLT equivalences given in \cite{Lee21, Mil19, LOR23}. 

\begin{thm}\label{thm: second main s-positivities} For a partition $\nu$, let $\mu,\lambda\subseteq \nu$ be two distinct partitions such that $|\nu/\lambda|=|\nu/\mu|=1$. Moreover, suppose $\nu/\lambda$ is a cell in the first or the second row. Then $\I_{\lambda,\mu}[X;q,t]$
positively expands in LLT polynomials. In particular, $\I_{\lambda,\mu}[X;q,t]$ is Schur positive.
\end{thm}

As a corollary, we prove Butler's conjecture for $q=1$ or $t=1$ (for general $\mu$ and $\lambda$).
\begin{corollary}\label{Cor: Schur positivity at t=1} Let $\lambda$ be a partition and $\mu$ be a partition obtained from $\lambda$ by moving a cell of a partition $\lambda$ to the upper row. Then $\I_{\lambda,\mu}[X;q,1]$ (or $\I_{\lambda,\mu}[X;1,q]$)
is Schur positive. 
\end{corollary}




\section{Combinatorial formulas for $(q,t)$-Kostka polynomials}\label{Sec: Schur expansions of Macodnald polynomials}

The modified $(q,t)$-Kostka polynomial $\widetilde{K}_{\mu,\nu}(q,t)$ is the Schur coefficient of the modified Macdonald polynomial: 
\[
    \widetilde{H}_\mu[X;q,t] = \sum_{\lambda}\widetilde{K}_{\mu,\lambda}(q,t)s_\lambda[X].
\]
This section provides combinatorial formulas for $(q,t)$-Kostka polynomials when $\mu$ is a two-column partition or a hook.

\subsection{Two-column case}\label{subsec: Two column}
For $n\ge 1$, we define a directed weighted graph $G(n)$ on 2-bounded partitions $p$ of size between $n$ and $2n$, where there is a directed edge from $p$ to $q$ if and only if $p\xRightarrow{}q$ is a strong marked cover. The weight of each edge is the spin statistic for each corresponding strong marked cover. Then one may think of vertical strong marked tableaux of $(2^n)$ as paths of the (hexagonal) graph, defined on 2-bounded partitions with edges corresponding to strong marked cover, from top to bottom. For a path $P$, we define the \emph{end partition} $\End(P)$ of $P$ as the 2-bounded partition corresponding to the end point of $P$ and \emph{weight} $\wt(P)$ of the path $P$ as the sum of weights of the edges in $P$. We now give a combinatorial formula for $2-$Schur expansion for the modified Macdonald polynomials of two-column partitions. The formula is motivated by the \emph{vertical dual Pieri rule}, and the \emph{shift invariance} of $k$-Schur functions studied by Blasiak, Morse, Pun, and Summers \cite{BMPS19}. While working on this paper, we found that our formula is equivalent to Zabrocki's formula \cite{Zab98}.

\begin{thm}\label{thm: combinatorial formula for two column}
Let $n$ and $m$ be positive integers with $n\ge 2m$. Then we have
\begin{equation}\label{eq: 2-Schur expansion formula for two column Macdonald polynomials}
    \omega\left(\widetilde{H}_{(2^m,1^{1^n-2m})}[X;q,t]\right) = \sum_{P} \wt(P)\prod_{i\in L(P)}\dfrac{q}{t^i}s^{(2)}_{\End(P)},
\end{equation}
where $\omega$ is the conjugate map on the symmetric function ring given by $h_n\rightarrow e_n$ for $n\ge 1$ and $s^{(2)}_\mu[X]$ is the 2-Schur function. In particular, this formula is consistent with Butler's conjecture.
\end{thm}

% As an example, Table~\ref{table: two columns of size 4} summarizes how the $2$-Schur coefficients of the (omega-image of) modified Macdonald polynomials of two-column partitions changes step by step while moving cells from $(1,1,1,1)$ to $(2,1,1)$, and to $(2,2)$.

% \begin{table}[h]
%     \centering
%     \begin{tabular}{|cc|cc|cc|c|}\hline
%     $\lambda$& \text{paths} & $(1,1,1,1)$ && $(2,1,1)$ && $(2,2)$ \\\hline
    
%     $(2,2)$ & $P_1$ & $t^4$ &$\rightarrow$& $q^3t$ &$\rightarrow$& $q^4$\\ \hline
    
    
%     $(2,1,1)$& $P_2$ & $t^3$ & $\rightarrow$ & $q$ && $q$\\
%     & $P_3$ & $t^2$ && $t^2$ &$\rightarrow$& $qt$\\ \hline
    
    
%     $(1,1,1,1)$ & $P_4$ & $1$ && $1$ && $1$\\ \hline
%     \end{tabular}
%     \caption{Table for 2-Schur coefficients of $\omega\left(\widetilde{H}_\mu[X;q,t]\right)$ for two-column partitions $\mu$.}
%     \label{table: two columns of size 4}
% \end{table} 


\subsection{Hook case}\label{subsec: hook}
In \cite{Ass18}, Assaf defined the bijections $\mathcal{A}_k$ for $k\ge 1$. Using these bijections, she gave a combinatorial formula for the Schur expansion of the modified Macdonald polynomials indexed by hook partitions. This formula is indeed consistent with Butler's conjecture. By showing this, we provide the following formula, which only involves the `ordinary' major statistic.

\begin{thm}\label{thm: formula for hook}
For $\mu=(n-k,1^k)$ a hook partition, we have
\[
\widetilde{H}_\mu[X;q,t] = \sum_{\lambda\vdash n}\sum_{w\in \SSS(\lambda)}\left(\prod_{\substack{1\le i \le k-1, \\ i \in \des(w) }}t^{\maj(w|_{[k]})}\prod_{\substack{k \le i \le n-1, \\ i \in \des(w)}}\dfrac{q^{n-i}}{t^i}\right)s_\lambda[X],
\]
where $\SSS(\lambda)$ is the set of superstandard tableaux of shape $\lambda$. In particular, this formula is consistent with Butler's conjecture.
\end{thm}

% As an example, Table~\ref{table: hooks of length 4} summarizes how $(q,t)$-statistics change step by step while moving cells from $(1,1,1,1)$ to get hook shapes of $4$.

% \begin{table}[ht]
%     \centering
%     \begin{tabular}{|ccc|cc|cc|cc|c|}\hline
%     $\lambda$& $\backslash$&$ \mu$ & $(1,1,1,1)$ && $(2,1,1)$ && $(3,1)$ && $(4)$ \\\hline
    
%     (4)&&$1234$ & 1 && 1 && 1 && 1 \\ \hline
    
    
%     $(3,1)$&&$12\dot{4}3$ & $t^3$ & $\rightarrow$ & $q$ && $q$ && $q$ \\
%     &&$1\dot{4}23$ & $t^2$ && $t^2$ & $\rightarrow$ & $q^2$ && $q^2$ \\
%     &&$\dot{4}123$ & $t$ && $t$ && $t$ & $\rightarrow$ & $q^3$\\ \hline
    
    
%     $(2,2)$ &&$\dot{3}1\dot{4}2$ & $t^4$ & $\rightarrow$ & $qt$ && $qt$ & $\rightarrow$ & $q^4$ \\ 
%     &&$3\dot{4}12$ & $t^2$ && $t^2$ & $\rightarrow$ & $q^2$ &&   $q^2$ \\ \hline
    
%     $(2,1,1)$ &&$1\dot{4}\dot{3}2$ & $t^5$ &$\rightarrow$& $qt^2$ &$\rightarrow$& $q^3$ && $q^3$ \\ 
%     &&$\dot{4}1\dot{3}2$ & $t^4$ &$\rightarrow$& $qt$ &&$qt$   &$\rightarrow$& $q^4$ \\ 
%     &&$\dot{4}\dot{3}12$ & $t^3$ && $t^3$ &$\rightarrow$& $q^2t$ &$\rightarrow$& $q^5$ \\ \hline
    
%     $(4)$&&$\dot{4}\dot{3}\dot{2}1$ & $t^6$ &$\rightarrow$& $qt^3$ &$\rightarrow$& $q^3t$ &$\rightarrow$& $q^6$ \\ \hline
%     \end{tabular}
%     \caption{Table for $\widetilde{K}_{\mu,\lambda}(q,t)$ for hook partitions $\mu$.}
%     \label{table: hooks of length 4}
% \end{table} 

\acknowledgements{The authors are grateful to François Bergeron, James Haglund, and Brendon Rhoades for helpful conversations. The authors also would like to thank Foster Tom, who shared the SAGE code for computing LLT polynomials.}

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