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%% define your title in the usual way
\title{Extending the Science Fiction and the Loehr--Warrington Formula}

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\author{\texorpdfstring{Donghyun Kim\thanks{\href{mailto:hyun920310@snu.ac.kr}{hyun920310@snu.ac.kr}. Donghyun Kim was partially 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} \and Jaeseong Oh \thanks{\href{mailto:jsoh@kias.re.kr}{jsoh@kias.re.kr}. Jaeseong Oh was partially supported by KIAS Individual Grant (HP083401) at  June E Huh Center for Mathematical Challenges in Korea Institute for Advanced Study.\addressmark{2}}}{Donghyun Kim and Jaeseong Oh }}

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\address{\addressmark{1}Department of Mathematical Sciences, Seoul National University, Seoul 151-247, Korea \\ \addressmark{2}June E Huh Center for Mathematical Challenges, Korea Institute for Advanced Study, Seoul 02455, Korea}

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\abstract{We introduce the Macdonald piece polynomial $\operatorname{I}_{\mu,\lambda,k}[X;q,t]$, which is a vast generalization of the Macdonald intersection polynomial in the science fiction conjecture by Bergeron and Garsia. We demonstrate a remarkable connection between $\operatorname{I}_{\mu,\lambda,k}$, $\nabla s_{\lambda}$, and the Loehr--Warrington formula $\operatorname{LW}_{\lambda}$, thereby obtaining the Loehr--Warrington conjecture as a corollary. We also present an extension of the science fiction conjecture and the Macdonald positivity by exploiting $\operatorname{I}_{\mu,\lambda,k}$.
}

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\keywords{Science fiction conjecture, Macdonald piece polynomials, Macdonald positivity, Loehr--Warrington conjecture, Garsia--Haiman--Tesler plethystic formula, \\Haglund--Haiman--Loehr formula, $P$-tableau}

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\begin{document}

\maketitle
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\section{Introduction}\label{Sec: Intro}
An important question posed by Macdonald was the \emph{Macdonald positivity conjecture} \cite{Mac88} which asserts that the \emph{modified Macdonald polynomial} $\widetilde{H}_\mu[X;q,t]$ is Schur positive. To address this conjecture, Garsia and Haiman introduced the bigraded \(\mathfrak{S}_n\) module \( R_\mu \), now called the \emph{Garsia--Haiman module} \cite{GH93}.

In his groundbreaking work \cite{Hai01}, Haiman utilized the geometry of the Hilbert scheme of \( n \) points in the plane \( \mathbb{C}^2 \) to prove that
\(
    \label{eq:Frob R_mu = H_mu}
    \mathrm{grFrob}\left( R_\mu; q, t \right) = \widetilde{H}_\mu[X; q, t],
\)
thereby settling the Macdonald positivity conjecture. Despite Haiman's resolution of Schur positivity, an explicit combinatorial formula for the Schur coefficients remains elusive.
\begin{problem}\label{prob: Macdonald-Kostka}
Find a combinatorial description for the Schur coefficient \(\widetilde{K}_{\lambda, \mu}(q, t)\) of the Macdonald polynomials:
\(
    \widetilde{H}_\mu[X;q,t] = \sum_{\lambda} \widetilde{K}_{\lambda, \mu}(q, t) s_\lambda,
\)
where \(s_\lambda\) is the Schur function.
\end{problem}

One approach to resolving Problem~\ref{prob: Macdonald-Kostka} is to decompose \(\widetilde{H}_{\mu}\) into Schur-positive pieces and find a combinatorial description of their Schur coefficients. The HHL formula \cite{HHL05} decomposes \(\widetilde{H}_{\mu}\) into LLT polynomials, which are Schur positive \cite{GH07}, though describing their Schur coefficients remains challenging. Another approach uses \(k\)-Schur functions \cite{LLM03}, with explicit Schur coefficients described in \cite{BMPS19}, but it's unclear if \(\widetilde{H}_{\mu}\) can be positively expressed in terms of \(k\)-Schur functions. In this paper, we propose an alternative method, generalizing the science fiction conjecture of Bergeron and Garsia \cite{BG99}.

Let $\mu$ be a partition with $n$ removable corners $\{c_1,\dots,c_n\}$ and let $1\le k \le n-1$. For a subset $S \in \binom{[n]}{k}$ of $[n]$ of size $k$, let $\mu^{S}$ be the partition obtained from $\mu$ by deleting corners $\{c_i : i \in S\}$. Let $R(n,k) := ((n-k)^k)$ be the rectangular partition, and let $\lambda$ be a partition inside $R(n,k)$. We define the \emph{Macdonald piece polynomial} $\I_{\mu,\lambda,k}$ as
\begin{equation}\label{eq: def of I_mu,lambda,k}
    \I_{\mu,\lambda,k}[X;q,t] := (-1)^{|\lambda|}\sum_{S \in \binom{[n]}{k}} \dfrac{s_\lambda[z_S] \prod_{j \in S^c}{z_j}}{\prod_{\substack{i \in S \\ j \in S^c}} (z_j - z_i)} \widetilde{H}_{\mu^S}[X].    
\end{equation}
Here, we let $z_i = q^{-\coarm(c_i)} t^{-\coleg(c_i)}$, and $f[z_S]$ denotes the polynomial $f$ in the $z_i$'s where $i$ runs through $S$. For $k=1$ and $\lambda=\emptyset$, $\I_{\mu,\lambda,k}$ recovers the Macdonald intersection polynomial $\I_{\mu^{(1)},\dots,\mu^{(n)}}$ defined in \cite{BG99} and studied in \cite{KLO22,KLO23}. Using the novel symmetric function \( \I_{\mu,\lambda,k} \), we define another symmetric function \( \widetilde{H}_{\mu^S}^{\lambda} \). We conjecture that it is the graded Frobenius characteristic of the intersection of certain Garsia--Haiman modules (see \Cref{conj: sf extend}), largely generalizing the science fiction conjecture.

For a partition \(\lambda \subseteq R(n,k)\), let \(\tilde{\lambda}\) be the conjugate of \((n-k-\lambda_k, \dots, n-k-\lambda_1)\), where \(\lambda_i = 0\) if \(i > \ell(\lambda)\). The notation \(\tilde{\lambda}\) depends on \(R(n,k)\), which will be clear from context. Now, we state our main results.

\begin{thm}\label{thm: main theorem} Let $\mu$ be a partition with $n$ corners, and $1\le k<n$. Then for a partition $\lambda$ inside a rectangle $R(n,k)$ the symmetric function $\I_{\mu,\lambda,k}$ satisfies the following:
\begin{enumerate}[label=(\alph*)]
    \item For $N>|\mu|-|\tilde{\lambda}|-k$, we have
    \(
        e_N^\perp \I_{\mu,\lambda,k} = 0.
    \)
    \item We have \(
        \dfrac{1}{T_{\mu^{[n]}}}e_{|\mu|-|\tilde{\lambda}|-k}^\perp \I_{\mu,\lambda,k} = \nabla s_{\tilde{\lambda}}.\)
    In particular, $e_{|\mu|+|\tilde{\lambda}|-k}^\perp \I_{\mu,\lambda,k}$ does not depend on the partition $\mu$ (up to a constant).
    \item We have \(
        \dfrac{1}{T_{\mu^{[n]}}}e_{|\mu|-|\tilde{\lambda}|-k}^\perp \I_{\mu,\lambda,k} = (-1)^{\adj(\tilde{\lambda})} \LW_{\tilde{\lambda}}.\)
\end{enumerate}
Here, $T_\mu := \prod_{c\in \mu}q^{\coarm(c)}t^{\coleg(c)}$, and definitions for $\LW_{\lambda}$ and $\adj(\lambda)$ is given in \Cref{sec: LW formula}.
\end{thm}
Note that the straightforward implication of \Cref{thm: main theorem} (b) and (c) is the Loehr--Warrington conjecture, which was very recently proved in \cite{BHMPS21LW}\footnote{There is also an unpublished paper by Carlsson and Mellit proving the Loehr--Warrington conjecture.
}.
\begin{corollary}[The Loehr--Warrington conjecture]\label{cor: LW thrm} 
    We have $\nabla s_{\lambda}=(-1)^{\adj(\lambda)}\LW_{\lambda}$.
\end{corollary}

\section{Extending the Science Fiction conjecture with Macdonald piece polynomials}
\label{sec: preliminaries extending the SF}
%  A partition $\mu=(\mu_1,\dots,\mu_\ell)$ is a weakly decreasing sequence of positive integers. We say that \(\mu\) is a partition of \(n\), denoted by \(\mu \vdash n\), if \(|\mu| := \sum_{i} \mu_i = n\). We denote the length of \(\mu\), i.e., the number of its parts, by \(\ell(\mu)\). We use \emph{Young diagrams} in French notation to display a partition as
% \[
%     \mu=\{(i,j)\in \mathbb{Z}_{\geq 1}\times \mathbb{Z}_{\geq 1} : j\le \mu_i\},
% \]
% where each element of this set represents a \emph{cell} in the diagram. The \emph{conjugate partition} of $\mu$, denoted by $\mu'=(\mu'_1,\mu'_2,\dots)$, is obtained by reflecting the Young diagram of $\mu$ along the diagonal $y=x$. We denote the empty partition by $\emptyset$.


% For a cell $u$ in a partition $\mu$, we define the \emph{arm} (\emph{coarm}) of $u$, denoted by $\arm_\mu(u)$ ($\coarm_\mu(u)$), as the number of cells strictly to the right (left) of $u$ in the same row. Similarly, we define the \emph{leg} (\emph{coleg}) of $u$, denoted by $\leg_\mu(u)$ ($\coleg_\mu(u)$), as the number of cells strictly above (below) $u$ in the same column. 

A \emph{diagram} is a collection of cells in the first quadrant. A partition can be viewed as a left-justified diagram with row lengths weakly decreasing from bottom to top. A cell in the \(i\)-th row and \(j\)-th column is denoted by \((i,j)\). We often represent a diagram as
\(
    D = [D^{(1)}, D^{(2)}, \dots],
\)
where \(D^{(j)} = \{i: (i,j) \in D\}\) is the set of row indices in the \(j\)-th column. The \emph{size} \( |D| \) is the number of cells in \( D \).

% $m_\lambda$ for the \emph{monomial symmetric function}, $h_\lambda$ for the \emph{homogeneous symmetric function}, $e_\lambda$ for the \emph{elementary symmetric function}, $p_\lambda$ for the \emph{power sum symmetric function}, $s_\lambda$ for the \emph{Schur function},  $\widetilde{H}_\lambda$ for the \emph{(modified) Macdonald polynomial}. The (Hall) inner product on symmetric functions, denoted by $\langle -,-\rangle$, is defined as \( \langle s_\lambda, s_\mu \rangle = \delta_{\lambda,\mu}.\)

We fix a partition $\mu$ with $n$ corners $\{c_1,c_2,\dots,c_n\}$ and let $\mu^{(i)}$ be a partition obtained by removing a corner $c_i$. Recall that the science fiction conjecture \cite{BG99} suggests that
\begin{equation}\label{eq: sf conjecture}
    \I_{\mu^{(1)},\dots,\mu^{(n)}}[X;q,t]:=\sum_{i=1}^{n}\left(\prod_{j\neq i}\dfrac{z_j}{z_j-z_i}\right) \widetilde{H}_{\mu^{(i)}}=\grFrob\left(\cap_{i=1}^{n}R_{\mu^{(i)}}\right),
\end{equation}
where $z_i=T_{\mu^{(i)}}/T_{\mu}$.
Note that $\I_{\mu^{(1)},\dots,\mu^{(n)}}[X;q,t]$ is a special case of the Macdonald piece polynomial $\I_{\mu,\lambda,k}$ obtained by letting $\lambda=\emptyset$ and $k=1$. Our goal is to further generalize \eqref{eq: sf conjecture} and refine the Macdonald positivity as well (\Cref{conj: sf extend}).

Note that $\I_{\mu,\lambda,k}$ can be regarded as a linear combination of $\widetilde{H}_{\mu^{S}}$. We present the converse by representing each $\widetilde{H}_{\mu^{S}}$ as a linear combination of $\I_{\mu,\lambda,k}$.

\begin{lem}\label{lem: H in terms of I}
We have 
\begin{equation*}
    \widetilde{H}_{\mu^{S}}=\sum_{\lambda\subseteq R(n,k)} \frac{s_{\tilde{\lambda}}[z_{S^{c}}]}{\prod_{j\in S^{c}}z_j}\I_{\mu,\lambda,k}
\end{equation*}
where $z_i = T_{\mu_{\{i\}}} / T_\mu = q^{-\coarm(c_i)} t^{-\coleg(c_i)}$.
\end{lem}

From now on we regard $z_1,z_2,\dots,z_n$ as indeterminates and let
\begin{equation*}
    \widetilde{H}_{\mu^{S}}[X;q,t,z]:=\sum_{\lambda\subseteq R(n,k)}\frac{s_{\tilde{\lambda}}[z_{S^{c}}]}{\prod_{j\in S^{c}}z_j}\I_{\mu,\lambda,k}.
\end{equation*}
Define $\Phi$ to be a specialization map on functions in $z_1,z_2,\dots,z_n$ given by letting $z_i = T_{\mu^{\{i\}}}/T_\mu$. By \Cref{lem: H in terms of I}, we trivially have $\Phi( \widetilde{H}_{\mu^{S}}[X;q,t,z])=\widetilde{H}_{\mu^{S}}$.

Let $\pi_{i,j}$\footnote{This operator is motivated by Butler's symmetric function \cite{But94} and for $j=i+1$, $\pi_{i,j}$ coincides with the \emph{Demazure operator $\pi_i$}.} be the operator acting on functions in $z_1,z_2,\dots,z_n$ defined by
\begin{equation*}
    \pi_{i,j}f(z_1,z_2,\cdots,z_n)=\frac{z_j f - z_i f\vert_{z_i \leftrightarrow z_j}}{z_j-z_i}.
\end{equation*}

Denoting $S=\{i_1<i_2<\dots<i_k\}$ and $S^{c}=\{j_1<j_2<\dots<j_{n-k}\}$, for a partition $\lambda \subseteq R(n,k)$ we define $\pi_{\lambda,S}$ to be a sequence of operators given by
\(   \pi_{\lambda,S}=\prod_{r=1}^{\ell(\lambda)}\prod_{s=1}^{\lambda_r}\pi_{i_r,j_s}.
\)
We may consider $\widetilde{H}_{\mu^{S}}[X;q,t,z]$ as a function in $z_1,z_2,\dots,z_n$ regarding $\I_{\mu,\lambda,k}$ as coefficients. In this sense the operators $\pi_{i,j}$ act on $ \widetilde{H}_{\mu^{S}}[X;q,t,z]$ by
\begin{equation*}
    \pi_{\lambda,S}( \widetilde{H}_{\mu^{S}}[X;q,t,z])=\sum_{\nu\subseteq R(n,k)} \left( \pi_{\lambda,S}\frac{s_{\tilde{\nu}}[z_{S^{c}}]}{\prod_{j\in S^{c}}z_j}\right)\I_{\mu,\nu,k}.
\end{equation*}
Finally, we define 
\begin{equation}\label{eq: module intersection sym def}
    \widetilde{H}^{\lambda}_{\mu^{S}}:=\Phi\left(\pi_{\lambda,S}( \widetilde{H}_{\mu^{S}}[X;q,t,z])\right).
\end{equation}
Now we give a conjectural module theoretic interpretation of $\widetilde{H}^{\lambda}_{\mu^{S}}$.
\begin{definition}
    For $S\in\binom{[n]}{k}$ and a partition $\lambda \in R(n,k)$, denote by \(
        S=\{i_1<i_2<\dots<i_k\}, S^{c}=\{j_1<j_2<\dots<j_{n-k}\}.
        \)
    We define $S(\lambda)$ to be a set of $S'\in\binom{[n]}{k}$ satisfying the following that if we denote \(
        S\setminus S'=\{i_{a_1}<i_{a_2}<\dots<i_{a_r}\}, S'\cap S^{c}=\{j_{b_1}>j_{b_2}>\dots>j_{b_r}\},\) then we have $b_{s}\leq \lambda_{a_s}$ for $1\leq s\leq r$.
\end{definition}
\begin{conj}\label{conj: sf extend} We have \(
\widetilde{H}^{\lambda}_{\mu^{S}}=\grFrob\left(\cap_{S'\in S(\lambda)}R_{\mu^{S'}}\right).
\)
\end{conj}
Letting $\lambda$ to be a full rectangle $R(n,k)$ the right-hand side is the Frobenius characteristic of the full intersection. Moreover, setting $k=1$ and $\lambda=R(n,1)$, \Cref{conj: sf extend} simply reduces to \eqref{eq: sf conjecture}.

For partitions $\lambda^{(1)}\subset \lambda^{(2)}$, we trivially have $S(\lambda^{(1)}) \subset S(\lambda^{(2)})$, and  \Cref{conj: sf extend} implies that \( \widetilde{H}^{\lambda^{(1)}}_{\mu^{S}}- \widetilde{H}^{\lambda^{(2)}}_{\mu^{S}}
\)
is Schur positive. Given a sequence $\emptyset=\lambda^{(1)}\subset \lambda^{(1)}\subset \dots \subset \lambda^{(\ell)}=R(n,k)$, we may obtain a sequence of symmetric functions that grows from $\widetilde{H}^{R(n,k)}_{\mu^{S}}$ to $\widetilde{H}^{\emptyset}_{\mu^{S}}=\widetilde{H}_{\mu^{S}}$ in a Schur positive sense. This refines the Macdonald positivity.








\section{Proof of \Cref{thm: main theorem} (a) and (b)}
\label{sec3: GHT}
\subsection{Technical Lemmas}

We begin by recalling several technical lemmas. To state the first lemma, we review the following notations:
\(
    M = (1-q)(1-t),
    B_\mu = \sum_{c\in\mu} q^{\coarm_\mu(c)} t^{\coleg_\mu(c)}, D_\mu = M B_\mu - 1, \tilde{h}_\mu = \prod_{c\in\mu}(q^{\arm_\mu(c)} - t^{\leg_\mu(c) + 1}), \tilde{h}'_\mu = \prod_{c\in\mu} (t^{\leg_\mu(c)} - q^{\arm_\mu(c) + 1}).
\)
We will use brackets to denote plethystic substitutions. Furthermore, for a symmetric function \( f \), define \( \Pi'_f \) as
\(
    \Pi'_f[X; q, t] = \nabla^{-1} f[X - \epsilon],
\)
and \( \rev(f) \) as the \( q, t \)-reversal
\(
    \rev(f) = f\big|_{q \mapsto q^{-1}, t \mapsto t^{-1}}.
\)
The following lemma \cite[Equation (3.4)]{KLO23} provides a formula for the Macdonald polynomial skewed by an elementary symmetric function. The proof of the lemma uses Garsia, Haiman, and Tesler's plethystic formula \cite[Theorem I.2]{GHT99}.
\begin{lem}[{\cite{KLO23}}]\label{lem: skewing Macdonald}
For a partition $\mu \vdash n$, we have
\[
    e^{\perp}_{n-m} \widetilde{H}_\mu = (qt)^m T_\mu \sum_{\lambda \vdash m} \rev \left( \Pi'_{\widetilde{H}_\lambda}[D_\mu; q, t] \right) \frac{T_\lambda \widetilde{H}_\lambda}{\tilde{h}_\lambda \tilde{h}'_\lambda}.
\]
\end{lem}
% \begin{proof}
%     Below, we give a sketch of a proof. 
    
%     Garsia, Haiman, and Tesler's plethystic formula \cite[Theorem I.2]{GHT99} states that for a symmetric function \( f \),
%     \[
%         \langle \omega f, e_{n-m}^\perp \omega \widetilde{H}_\mu \rangle_* = \Pi'_f[D_\mu].
%     \]
%     By setting \( f = \widetilde{H}_\lambda \) in this equation and utilizing \eqref{eq: reversal *-inner product} along with the symmetry relation \( \omega \widetilde{H}_\mu = T_\mu \rev(\widetilde{H}_\mu) \), we obtain
%     \[
%         \frac{1}{(qt)^m T_\lambda T_\mu} \langle \widetilde{H}_\lambda, e_{n-m}^\perp \widetilde{H}_\mu \rangle = \rev \left( \Pi'_{\widetilde{H}_\lambda}[D_\mu; q, t] \right).
%     \]
%     Applying the orthogonality relation \eqref{Eq: orthogonality of Mac w.r.t <>*}, we conclude the proof.
% \end{proof}
We need one more lemma, which follows from the Laplace expansion of the determinant using complementary minors and Jacobi's bi-alternant formula for Schur functions.
\begin{lem}\label{lem: Schur orthogonal in rectangle}
Let \( z_1, \dots, z_n \) be variables. Let \( \lambda \) be a partition contained within the rectangle \( R(n,k) \). For a partition \( \mu \) of size \( |\mu| \le k(n-k) - |\lambda| \),
\begin{equation}\label{eq: Schur orthogonal}
    \sum_{S \in \binom{[n]}{k}} \frac{s_\lambda[z_S] s_{\mu}[z_{S^c}]}{\prod_{i \in S, j \in S^c} (z_j - z_i)} = (-1)^{|\lambda|}\delta_{\tilde{\lambda}, \mu}.
\end{equation}
\end{lem}

\subsection{Proof of \Cref{thm: main theorem} (a) and (b)}

Let \(\mu\) be a partition of \(N\) with \(n\) corners \(\{c_1, \dots, c_n\}\). Fix \(0 \le k \le n\) and let \(\lambda\) be a partition inside the rectangular partition $R(n,k)$. Let \(m = k(n-k) - |\lambda|=|\tilde{\lambda}|\). By the definition of \(\I_{\mu,\lambda,k}\), for an integer \(\ell\),
\begin{equation}
    e^\perp_{N-k-\ell} \I_{\mu,\lambda,k} = \sum_{S \in \binom{[n]}{k}} \frac{s_\lambda[z_S] \prod_{j \in S^c} z_j}{\prod_{i \in S, j \in S^c} (z_j - z_i)} e^\perp_{N-k-\ell} \widetilde{H}_{\mu^S}.
\end{equation}
By \Cref{lem: skewing Macdonald}, we have
\begin{equation}\label{eq: skewing I}
    e^\perp_{N-k-\ell} \I_{\mu,\lambda,k} = (qt)^\ell T_{\mu^{[n]}} \sum_{\nu \vdash \ell} \sum_{S \in \binom{[n]}{k}} \frac{s_\lambda[z_S]}{\prod_{i \in S, j \in S^c} (z_j - z_i)} \rev(\Pi_{\widetilde{H}_\nu}' [D_{\mu^S}(q,t); q, t]) \frac{T_\nu \widetilde{H}_\nu}{\tilde{h}_\nu \tilde{h}_\nu'}.
\end{equation}
Here, we used \(T_{\mu^{[n]}}= T_{\mu^S} \prod_{j \in S^c} z_j\). Note that 
\[
    \rev\left(\Pi'_{\widetilde{H}_\nu}[D_{\mu^S}(q,t); q, t]\right) = T_\nu \widetilde{H}_\nu \left[D_{\mu^{[n]}}(q^{-1}, t^{-1}) - \epsilon + \frac{M}{qt} z_{S^c}; q^{-1}, t^{-1}\right]
\]
is a polynomial in \(z_{S^c}\) of degree \(|\nu| = \ell\). If, on the right-hand side of \eqref{eq: skewing I}, the degree of the polynomial in \(z_{S^c}\) is less than \(m\), or equivalently if \(\ell < m\), then by \Cref{lem: Schur orthogonal in rectangle}, it vanishes. This proves \Cref{thm: main theorem}~(a).


Proceed to prove \Cref{thm: main theorem} (b). Now let \(\ell = m\) in \eqref{eq: skewing I}. Note that the leading term of \(\rev\left(\Pi'_{\widetilde{H}_\nu}[D_{\mu^S}(q,t); q, t]\right)\) is equal to \(
T_\nu \widetilde{H}_\nu \left[\frac{M}{qt} z_{S^c}; q^{-1}, t^{-1}\right] = (qt)^{-m}\omega \widetilde{H}_\nu \left[M z_{S^c}; q, t\right],\)
by the symmetry relation $\omega \widetilde{H}_\mu = T_\mu \rev(\widetilde{H}_\mu)$.
Again, by \Cref{lem: Schur orthogonal in rectangle}, we only need to consider this leading term. Let us rewrite the right-hand side of \eqref{eq: skewing I} as
\begin{align*}
    & T_{\mu^{[n]}} \nabla \left(\sum_{\nu \vdash \ell} \sum_{S \in \binom{[n]}{k}} \frac{s_\lambda[z_S]}{\prod_{i \in S, j \in S^c} (z_j - z_i)} 
    \frac{\widetilde{H}_\nu \omega \widetilde{H}_\nu \left[M z_{S^c}; q, t\right]}{\tilde{h}_\nu \tilde{h}_\nu'}\right) \\ & \quad = T_{\mu^{[n]}} \nabla \left(\sum_{\rho \vdash \ell } \sum_{S \in \binom{[n]}{k}} \frac{s_\lambda[z_S] s_{\rho'}[z_{S^c}]}{\prod_{i \in S, j \in S^c} (z_j - z_i)} s_{\rho'}\right),
\end{align*}
where we used \cite[(2.67)]{Hag04}:
\(
    s_{\rho'} = \sum_{\nu\vdash |\rho|} \dfrac{\tilde{K}_{\rho,\nu}(q,t) \widetilde{H}_{\nu}[MX;q,t]}{\tilde{h}_\nu \tilde{h}'_\nu}.
\) 
Now, applying \Cref{lem: Schur orthogonal in rectangle} again we obtain 
\(
T_{\mu^{[n]}} \nabla \left(\sum_{\rho \vdash \ell} \sum_{S \in \binom{[n]}{k}} \frac{s_\lambda[z_S] s_{\rho'}[z_{S^c}]}{\prod_{i \in S, j \in S^c} (z_j - z_i)} s_{\rho'}\right) = T_{\mu^{[n]}} \nabla s_{\tilde{\lambda}}.
\)


\section{Loehr--Warrington formula}

\label{sec: LW formula}
\subsection{The Loehr--Warrington formula} \label{subsec: LW formula}
Throughout this section, we fix a poset $\PP$ on $\mathbb{Z}_{\ge 0}\times \mathbb{Z}_{\ge1}$ defined as follows. For $(a,b)$, $(c,d) \in \mathbb{Z}_{\geq 0}\times \mathbb{Z}_{\geq 1}$ we say $(a,b) \prec_{\PP} (c,d)$ in $\PP$ if and only if $a + 1 < c$ or $a + 1 = c$ and $b \ge d$. Otherwise, we write $(a,b) \nprec_{\PP} (c,d)$. We also give a total ordering on $\mathbb{Z}_{\geq 0}\times \mathbb{Z}_{\geq 1}$ by a (mixed) lexicographic ordering. Define $(a,b)<_{\lex}(c,d)$ if $a<c$, or $a=c$ and $b>d$. Note that the ordering of the second coordinate is reversed.

Consider a tuple $L=(L_1,\dots,L_r)$ such that each $L_i$ is a finite sub(multi)set of $\mathbb{Z}_{\geq0} \times \mathbb{Z}_{\geq1}$. We define the \emph{diagonal inversion} $\dinv$ of $L$ by 
\begin{align*}
    \dinv(L)&=\sum_{i<j}\sum_{\substack{(a,b)\in L_i\\ (a',b')\in L_j}}\chi((a,b)<_{\lex}(a',b'))\chi((a,b)\nprec_{\PP}(a',b')).
    \end{align*}
For a diagram \( D \), let \( T \) be a filling of \( D \) with elements from \(\mathbb{Z}_{\geq 0} \times \mathbb{Z}_{\geq 1} \). Denote \( T(i,j) \) to be a filling in the cell \((i,j) \in D\). We use \( T(i,j)_1 \) (resp. \( T(i,j)_2 \)) to refer to the first (resp. second) entry of \( T(i,j) \). We define the area of \( T \) as
\(
\area(T) = \sum_{(i,j) \in D} T(i,j)_1.
\)
We use the notation \(\dinv(T)\) to denote \(\dinv(L_1, \dots, L_r)\), where \( L_i \) represents the set of entries in the \( i \)-th column of \( T \).

We fix a partition $\lambda \subseteq R(n,k)$. Let $s$ be the size of the \emph{Durfee square}, which is the maximal number such that $\lambda_s\geq s$. We define a \textit{dinv adjustment} of $\lambda$ denoted by $\adj(\lambda)$ as
\(
\adj(\lambda)=\sum_{i=1}^{s}(\lambda_i-i).
\)
Consider a $n$-vector
\(
    (\lambda_1+k-1,\lambda_2+k-2,\dots,\lambda_k,k,k+1,\dots,n-1)
\)
where we regard $\lambda_i=0$ if $i>\ell(\lambda)$, and define $v(\lambda)$ to be a vector obtained by sorting entries in the above vector in a weakly increasing order. Then we define the \textit{bottom} of $\lambda$, denoted by $\bo(\lambda)$, as an $n$-vector given by
\(
\bo(\lambda) = (s+1, s+2, \ldots, s+n) - v(\lambda),
\)
where the subtraction is performed element-wise. We also define a \textit{pivot} of $\lambda$, denoted by $\piv(\lambda)$, to be a vector $(a_1,a_2,\dots,a_s)$ of length $s$ where $a_i$ is a number satisfying
\begin{equation*}
    v(\lambda)_{a_i}=v(\lambda)_{a_i+1}=\lambda_{s+1-i}+k-(s+1-i),
\end{equation*}
i.e., the indices where $v(\lambda)$ is non increasing. Lastly we associate a diagram $D(\lambda)$ to the partition $\lambda$ as
\begin{equation*}
    D(\lambda)=[[\bo(\lambda)_1,s],[\bo(\lambda)_2,s],\dots,[\bo(\lambda)_n,s]]
\end{equation*}
and define a set $\mathcal{T}(\lambda)$ consisting of filling $T$ of $D(\lambda)$ satisfying the following conditions: 
\begin{itemize}
    \item $T(i+1,j)\succ_{\PP} T(i,j)$ and $T(i,j-1)\nsucc _{\PP} T(i,j)$
    \item for each $j> k-s$ if $j\in \piv(\lambda)$ we have $T(\bo(\lambda)_j,j)_1>0$
    \item for each $j> k-s$ if $j\notin \piv(\lambda)$ we have $T(\bo(\lambda)_j,j)_1=0$.
\end{itemize}

Note that the first condition says that $T$ (after taking appropriate reflection) is a $\PP$-tableau defined in \cite{Gas96}. 
The \emph{Loehr--Warrington formula} $\LW_{\lambda}$ in \cite{LW08} can be rewritten by
\begin{equation}\label{eq: LW formula}
\LW_{\lambda}:=q^{\adj(\lambda)} \sum_{T\in \mathcal{T}(\lambda)}q^{\dinv(T)}t^{\area(T)}x^T, \qquad \text{where } x^T=\prod x_{T(i,j)_2}
\end{equation}

\subsection{Jacobi--Trudi type formula for the Loehr--Warrington formula} 
In this section, we provide a Jacobi--Trudi type formula (\Cref{prop: JT formula}) for $\LW_{\lambda}$ in terms of the operators $\mathfrak{h}_m, \bar{\mathfrak{h}}_m$, and $\hat{\mathfrak{h}}_m$, defined below.
\begin{definition}\label{def: operator definition}
We define $\mathbf{C}_m$ to be a set of all $\PP$-chains $\{(a_1,b_1) \prec_{\PP} (a_2,b_2)\prec_{\PP} \dots \prec_{\PP} (a_m,b_m)\}$ where each $(a_\ell,b_\ell) \in \mathbb{Z}_{\geq0} \times \mathbb{Z}_{\geq1}$. Then $\bar{\mathbf{C}}_m$ (resp $\hat{\mathbf{C}}_m$) is a subset of $\mathbf{C}_m$ consisting of elements satisfying $a_1=0$ (resp $a_1>0$). Obviously, $\mathbf{C}_m=\bar{\mathbf{C}}_m \cup\mkern-11.5mu\cdot\mkern5mu \hspace{0.05cm}\hat{\mathbf{C}}_m$.

Let $\textbf{y}=\{y_{i,j}\}_{i \in \mathbb{Z}_{\geq 0}, j \in \mathbb{Z}_{\geq 1}}$ be a set of indeterminates. For a multiset $A$ of elements in $\mathbb{Z}_{\geq 0} \times \mathbb{Z}_{\geq 1}$, let $y^A$ be the monomial whose exponent of $y_{i,j}$ is given by the number of elements $(i,j)$ in $A$. We define the operators $\mathfrak{h}_m$, $\bar{\mathfrak{h}}_m$, and $\hat{\mathfrak{h}}_m$ acting on a polynomial ring $\mathbb{F}[\textbf{y}]$ ($\mathbb{F}$ is a ground field containing $\mathbb{C}(q)$) by describing their actions on a monomial as
\[
\mathfrak{h}_m \cdot y^A = \sum_{L \in \mathbf{C}_m} q^{\dinv(L,A)} y^{(L,A)}, \quad
\bar{\mathfrak{h}}_m \cdot y^A = \sum_{L \in \bar{\mathbf{C}}_m} q^{\dinv(L,A)} y^{(L,A)}, \]
\[ \hat{\mathfrak{h}}_m \cdot y^A = \sum_{L \in \hat{\mathbf{C}}_m} q^{\dinv(L,A)} y^{(L,A)},\]
where $y^{(L,A)}=y^L y^A$. If $m<0$ then they are all zero operators. 
\end{definition}

\begin{lem}\label{lem: SW involution} We have $\mathfrak{h}_n \mathfrak{h}_m=\mathfrak{h}_m \mathfrak{h}_n$, $\bar{\mathfrak{h}}_n \bar{\mathfrak{h}}_m=\bar{\mathfrak{h}}_m \bar{\mathfrak{h}}_n$, and $\hat{\mathfrak{h}}_n \hat{\mathfrak{h}}_m=\hat{\mathfrak{h}}_m \hat{\mathfrak{h}}_n$
\end{lem}
\begin{proof}
The proof parallels with \cite[Lemma 4.4, Theorem 4.5]{SW16}.
\end{proof}

For an $n$ by $n$ matrix $W=(W_{i,j})$ whose entry is an operator acting on $\mathbb{F}[\textbf{y}]$, we define \(\det(W):=\sum_{\sigma\in \mathfrak{S}_n}(-1)^{\sgn(\sigma)}W_{\sigma(1),1}W_{\sigma(2),2}\dots W_{\sigma(n),n}. \)

\begin{proposition}\label{prop: JT formula}
For integers $k<n$ and a partition $\lambda \subseteq R(n,k)$ we associate an $n$ by $n$ matrix
\begin{align*}
    W(\lambda)_{i,j}=\begin{cases*}
        \mathfrak{h}_{v(\lambda)_j-i+1} \qquad \text{if $j\le k-s$}\\
        \bar{\mathfrak{h}}_{v(\lambda)_j-i+1} \qquad \text{if $j> k-s$ and $j$ is an entry of $\piv(\lambda)$}\\
        \hat{\mathfrak{h}}_{v(\lambda)_j-i+1} \qquad \text{if $j> k-s$ and $j$ is not an entry of $\piv(\lambda)$},
    \end{cases*}
\end{align*}
where $s$ is the size of the Durfee square of $\lambda$. Then we have 
\(
    q^{-\adj(\lambda)} \LW_\lambda = \det(W(\lambda)) \cdot 1 \vert_{y_{i,j}=t^i x_j}
\)
\end{proposition}


\subsection{Reformulation of Loehr--Warrington formula}
We reformulate the Loehr--Warrington formula \(\LW_{\lambda}\) from \Cref{prop: JT formula}. Our main tool is \Cref{lem: multiplication by q}, whose proof is given in the full version \cite{KO24} of this paper. For a matrix \( A \), let \( T_{ij}(A) \) be the matrix obtained by moving the \( j \)-th column to the position of the \( i \)-th column and shifting the \( \ell \)-th column to the \( (\ell+1) \)-th position for \( i < \ell < j \). 

% For example, we have
% \begin{equation*}
% A = \begin{pmatrix}
%     \vertbar & \vertbar &   \vertbar &  \vertbar  & \vertbar \\
%     c_{1}    & c_{2}    &    c_3    & c_{4}  & c_5  \\
%     \vertbar & \vertbar &    \vertbar &  \vertbar &\vertbar
%   \end{pmatrix}
% \longrightarrow T_{25} A = \begin{pmatrix}
%     \vertbar & \vertbar &   \vertbar &  \vertbar  & \vertbar \\
%     c_{1}    & c_{5}    &    c_2    & c_{3}  & c_4  \\
%     \vertbar & \vertbar &    \vertbar &  \vertbar &\vertbar
%   \end{pmatrix}.
% \end{equation*}


\begin{lem}\label{lem: multiplication by q}
For an integer vector \( v = (v_1, v_2, \dots, v_n) \) of length \( n \geq 2 \), consider an \( n \times n \) matrix \( V = (V_{i,j}) \) given by
\(
V_{i,j} = \begin{cases*}
\hat{\mathfrak{h}}_{v_i + j - 1} \quad \text{if \( j < n \)} \\
\bar{\mathfrak{h}}_{v_i + n - 1} \quad \text{if \( j = n \)}
\end{cases*}.
\)
Then we have \( (-q)^{n-1} \det V = \det T_{1,n}(V) \).
\end{lem}

Let \( s \) be the largest integer such that \(\lambda_s \geq s\). Set \( W^{(0)} = W(\lambda) \) as in \Cref{prop: JT formula}, and construct new matrices recursively as follows: 

(Step 1) Given \( W^{(i)} \), let \( V^{(i+1)} = T_{ab}(W^{(i)}) \) where \( a = k - s + i \) and \( b = \piv(\lambda)_i \). 

(Step 2) Add the \((\piv(\lambda)_i + 1)\)-th column to the \((k - s + i)\)-th column in \( V^{(i+1)} \) to obtain \( W^{(i+1)} \).

By \Cref{lem: multiplication by q}, We conclude that
\(
    (-q)^{\adj(\lambda)}\det(W(\lambda))=\det(W^{(s)}),
\)
and
\begin{equation}\label{eq: Wr matrix formula}
    W^{(s)}_{i,j} = \begin{cases*}
        \mathfrak{h}_{\lambda_{k+1-j} + j - i} \quad \text{if \( j \leq k \)} \\
        \hat{\mathfrak{h}}_{j - i} \quad \text{if \( j > k \)}.
    \end{cases*}
\end{equation}
Finally, by \Cref{prop: JT formula}, we conclude
\begin{equation}\label{eq: lw formula final}
    \LW_{\lambda} = (-1)^{\adj(\lambda)}\det(W^{(s)}) \cdot 1 |_{y_{i,j} = t^i x_j}.
\end{equation}

\section{Proof of \Cref{thm: main theorem} (c)}
\label{sec: deform}
% We outline the proof strategy:
% \begin{itemize}
%     \item We define auxiliary filled diagrams $(\delta_{S},f_{S})$, $(\delta_{S},f^{z}_{S})$ and $(\Rec^{\leq m},g^{z}_S)$ 
%     \item We go over a technical process to obtain \eqref{eq: rec final} from $\left(e_{|\delta|-|\lambda|-k}^\perp \I_{\delta,\tilde{\lambda},k}\right)$, imitating arguments in \cite[Section 6]{KLO23}
%     \item Then we illustrate a clear connection between \eqref{eq: rec final} and (reformulated) Jacobi--Trudi type formula \eqref{eq: lw formula final} for the Loehr--Warrington formula $\LW_\lambda$.
% \end{itemize}


\subsection{Macdonald polynomials for filled diagrams and HHL formula}\label{subsec: generalization of Macdonald}

To deal with the combinatorics of Macdonald piece polynomial, we rely on the celebrated result of Haglund--Haiman--Loehr \cite{HHL05,HHL08}. For a diagram $D$, and cells $u=(i,j)$ and $v=(i',j')$ of $D$, we say that a pair $(u,v)$ is an \emph{attacking pair} if either 

    \qquad \qquad \qquad \quad(1) $i=i'$ and $j<j'$ $\quad$ or \quad (2) $i=i'+1$ and $j>j'$.

For a word $\sigma \in\mathbb{Z}_{\geq 1}^{|D|}$, we consider a pair $(u,v)$ of cells in $D$ to be an \emph{inversion pair} of $\sigma$ if $(u,v)$ is an attacking pair and $\sigma(u)>\sigma(v)$. We define
\(
    \inv_D(\sigma) \coloneqq q^{|\Inv_D(\sigma)|},
\)
where $\Inv_D(\sigma)$ is the set of inversion pairs of $\sigma$.
A cell $u$ is a \emph{descent} of $\sigma$ if $\sigma(u)>\sigma(v)$, where the cell $v$ is the cell right below $u$, i.e. $u=(i,j)$ and $v=(i-1,j)$. Define $\Des_D(\sigma)$ to be the set of descents of $\sigma$.

A \emph{filling} of $D$ is a function $f:D\rightarrow \mathbb{F}$, where we take $\mathbb{F}$ as a field containing $\mathbb{C}(q,t)$. Then a \emph{filled diagram} $(D,f)$ is a pair of a diagram and a filling on it. We define $\maj_{(D,f)}(\sigma)$ as the product of $f(u)$ over all cells $u$ which are descents of $\sigma$, i.e.,
\(
    \maj_{(D,f)}(\sigma) \coloneqq \prod_{u\in\Des_D(\sigma)} f(u).
\)
Finally, $\stat_{(D,f)}(w)$ is defined by the product of $\inv_D$ and $\maj_{(D,f)}$,
\(
    \stat_{(D,f)}(\sigma) \coloneqq \inv_D(\sigma)\maj_{(D,f)}(\sigma).
\) In \cite{KLO22}, the \emph{(generalized) modified Macdonald polynomial} of a filled diagram $(D,f)$ is defined by
\(
    \widetilde{H}_{(D,f)} := \sum_{\sigma \in \mathbb{Z}_{\geq 1}^{|D|}} \stat_{(D,f)}(\sigma) x^{\sigma}.
\)
Indeed, this generalizes the celebrated HHL formula \cite{HHL05}, which gives the combinatorial formula for the modified Macdonald polynomial.


\subsection{Deformation of filled diagrams}\label{sub: 6.1}
By the shape independence in \Cref{thm: main theorem} (b), to prove \Cref{thm: main theorem} (c), it suffices to consider a specific partition \(\mu\) of \(n\) corners.  We fix \(\mu\) to be the $\delta=\delta_{n,N} \coloneqq (n^N,n-1,\dots,1)$ for a large enough $N$ ($N>|\lambda|$ suffices). 

Let the corners of $\delta$ be indexed by \(c_1, c_2, \dots, c_n\) from top to bottom. Additionally, for a $k$-subset $S=\{i_1<i_2<\cdots<i_k\}$ of \([n]\), let \(\delta^{S}\) denote the partition obtained from $\delta$ by deleting the corners $c_i$'s for $i
\in S$. We rearrange columns of $\delta^{S}$ by moving the $i_1, i_{2}, \dots, i_k$-th columns all the way to the left in this order and denote by $\beta$. We let $f^{\st}_\beta$ be the `standard filling' in \cite{HHL08}, or the filling obtained by the column exchange rule \cite[Proposition 5.3]{KLO22}. Now we define $(\delta_S,f_S):=\cycling^{k}(\beta,f^{\st}_{\beta})$ (the diagram $\delta_S$ should not be confused with $\delta^{S}$). Here $\cycling$ is the cycling operator (cf. \cite[Lemma 3.6]{KLO22}). We may use $(\delta_S,f_S)$ for the computation of $\widetilde{H}_{\delta^{S}}$ i.e.  $\widetilde{H}_{\delta^{S}}= \widetilde{H}_{(\delta_S,f_S)}$. 

Now, we introduce indeterminates $z_1,z_2,\dots,$ and define $(\delta_S,f^{z}_S)$ which is a $z$-de\-for\-ma\-tion of $(\delta_S,f_S)$. Let $S^{c}=\{j_1<j_2<\dots<j_{n-k}\}$. Then the filling $f^{z}_{S}$ is given by 
\[
    f^{z}_S(a,b)=q\frac{z_{N+n+1-a}}{z_{j_b}} \quad \text{for  $b\leq n-k$,} \]\[ f^{z}_S(a,n-k+b)=\begin{cases*}
        \frac{z_{N+n+1-a}}{z_{i_b}}\qquad \text{if $N+n+1-a\in S^{c}$ }\\
        q\frac{z_{N+n+1-a}}{z_{i_b}}\qquad \text{otherwise}
    \end{cases*} \quad \text{for $b\leq k$}.
\]
A straightforward calculation shows that under the specialization given by
\begin{equation}\label{eq: z_i values}
    z_j = 
    \begin{cases}
        q^{1-j}t^{j+1-N-n}&\text{if } j\le n\\
        q^{-n}t^{j+1-N-n}&\text{if } j > n
    \end{cases}
\end{equation}
$(\delta_S,f^{z}_S)$ recovers $(\delta_S,f_S)$. Lastly for a positive integer $m\leq N$, we define $(\Rec^{\leq m},g_{S}^{z})$ to be a sub filled diagram of $(\delta_S,f^{z}_S)$ obtained by restricting to the first $m$ rows. Note that the diagram $\Rec^{\leq m}$ is independent of $S$.
% In \cref{fig: delta S process}, examples of $(\delta^S,f^{\st}_S)$, $(\delta_S,f_S)$, $(\delta_S,f_S^z)$ and $(\Rec^{\le m}, g_S)$ are given.
% \ytableausetup
% {boxsize=2.5em}
%  \begin{figure}[ht] 
%  \qquad\qquad
% \scalebox{0.6}{\begin{ytableau}
%  \textcolor{red}{t} \\
%  \textcolor{red}{t^2} \\
%  \textcolor{red}{q^{-1}t^3} & \textcolor{blue}{t}\\
%  \textcolor{red}{q^{-3}t^4} & \textcolor{blue}{q^{-2}t^2} & \textcolor{blue}{q^{-1}t} &  \textcolor{red}{t}\\
%  \textcolor{red}{q^{-3}t^5} & \textcolor{blue}{q^{-2}t^3} & \textcolor{blue}{q^{-1}t^2} & \textcolor{red}{t^2}\\&&&\\
% \end{ytableau}}
% \qquad
%  \scalebox{0.6}{
%  \begin{ytableau}
%  \textcolor{red}{t} \\
%  \textcolor{red}{q^{-1}t^2} & \none & \textcolor{blue}{t} \\
%  \textcolor{red}{q^{-2}t^3} & \none & \textcolor{blue}{q^{-2}t^2} &  \textcolor{blue}{q^{-1}t}\\
%  \textcolor{red}{q^{-3}t^4} &  \textcolor{red}{t} & 
%  \textcolor{blue}{q^{-2}t^3} & \textcolor{blue}{q^{-1}t^2} \\
%  \textcolor{red}{q^{-3}t^5} & 
%  \textcolor{red}{t^2} & & \\ &
%  \end{ytableau}
%  }
%  \qquad
% \scalebox{0.6}{\begin{ytableau}
%  \textcolor{red}{q\dfrac{z_2}{z_1}} \\
%  \textcolor{red}{q\dfrac{z_3}{z_1}} & \none & \textcolor{blue}{q\dfrac{z_3}{z_2}} \\
%  \textcolor{red}{q\dfrac{z_4}{z_1}} & \none & \textcolor{blue}{\dfrac{z_4}{z_2}} &  \textcolor{blue}{\dfrac{z_4}{z_3}}\\
%  \textcolor{red}{q\dfrac{z_5}{z_1}} &  \textcolor{red}{q\dfrac{z_5}{z_4}} & \textcolor{blue}{q\dfrac{z_5}{z_2}} & \textcolor{blue}{q\dfrac{z_5}{z_3}} \\
%  \textcolor{red}{q\dfrac{z_6}{z_1}} & \textcolor{red}{q\dfrac{z_6}{z_4}} & & \\  &  \\
%  \end{ytableau}
%  }
%  \qquad
%   \scalebox{0.6}{ \begin{ytableau}
%      \textcolor{red}{q\dfrac{z_5}{z_1}} &  \textcolor{red}{q\dfrac{z_5}{z_4}} & \textcolor{blue}{q\dfrac{z_5}{z_2}} & \textcolor{blue}{q\dfrac{z_5}{z_3}} \\
%      \textcolor{red}{q\dfrac{z_6}{z_1}} & \textcolor{red}{q\dfrac{z_6}{z_4}} & & \\  &  \\
%      \end{ytableau}}
%  \caption{$(\delta^S,f^{\st}_S)$, $(\delta_S,f_S)$, $(\delta_S,f_S^z)$ and $(\Rec^{\le m}, g_S)$}
%  \label{fig: delta S process}
%  \end{figure}


\subsection{Reducing to the \texorpdfstring{$\Rec^{\leq N}$}{Req <= N}}\label{sub 6.2}

In this section we compute $\left(e_{|\delta|-k-|\lambda|}^\perp \I_{\delta,\tilde{\lambda},k}\right)$ starting from  filled diagrams $(\delta_S,f_{S})$ for the computation of $\widetilde{H}_{\delta^{S}}$. 
Given a diagram $D$ and a positive integer $\ell \leq |D|$, we define $\Sub(D,\ell)$ to be the set of assignments of $\ell$ cells of $D$ by positive integers. We think of $R\in \Sub(D,\ell)$ as a function $R: D'\rightarrow \mathbb{Z}_{\geq 1}$ for some subdiagram $D'\subseteq D$ with $|D'|=\ell$. We also define an induced full assignment $\bar{R}:D \rightarrow \mathbb{Z}_{\geq 1}$ given by assigning $1,2,\dots,|D|-\ell$ for cells in $D\setminus D'$ in a reverse reading order, and by letting $\bar{R}(c)=R(c)+|D|-\ell$ for cells $c\in D'$. We let $x^{R}=\prod_{c\in D'}x_{R(c)}$ and lastly for a filled diagram $(D,f)$, we abuse the notation for $\stat_{(D,f)}(R)$ instead of $\stat_{(D,f)}(\bar{R})$.

Let $D=\Rec^{\leq m}$ for some $m\leq N$ and consider $R\in \Sub(D,\ell)$. We say that a cell $c=(i,j)\in D'$ is \emph{bad} if $c'=(i+1,j)\in \Des_{D}(\bar{R})$ where $\bar{R}$ is the associated full assignments. We define $\gamma(R)$ to be a vector whose $i$-th entry is the number of bad cells in the $i$-th row. By employing delicate arguments, as outlined in the full version of this paper \cite[Section 6.2]{KO24}, one can show the following.
\begin{lem}\label{lem: final rec} 
    The following quantity specializes to $e_{|\delta|-k-|\lambda|}^\perp \I_{\delta,\tilde{\lambda},k}$ under the specialization \eqref{eq: z_i values}.
    \begin{equation}\label{eq: rec final}
    (-1)^{|\tilde{\lambda}|}\frac{\stat_{(\delta_S,f^{z}_{S})}(R)}{\stat_{(\Rec^{\leq N},g^{z}_{S})}(R)} \sum_{S \in \binom{[n]}{k}} \dfrac{s_{\tilde{\lambda}}[z_S] \prod_{j \in S^c}{z_j}}{\prod_{\substack{i \in S \\ j \in S^c}} (z_j - z_i)} \sum_{\substack{R\in \Sub(\Rec^{\leq N},|\lambda|)\\ \gamma(R)=0}}\stat_{(\Rec^{\leq N},g^{z}_{S})}(R)x^{R}
\end{equation}
\end{lem}
\subsection{Toward the Jacobi--Trudi type formula}\label{sub: 6.3}
\begin{lem}\label{lem: key connection JT formula} Let $\lambda\subseteq R(n,k)$, and $P(z_1,\dots,z_n)$ be a polynomial of degree $|\lambda|$ that is symmetric in variables $z_1,z_2,\dots,z_{n-k}$ and symmetric in variables $z_{n-k+1},z_{n-k+2},\dots,z_{n}$. Then we have 
\begin{equation*}
     \sum_{S \in \binom{[n]}{k}} \dfrac{s_{\tilde{\lambda}}[z_S]P(z_{S^{c}},z_S)}{\prod_{\substack{i \in S \\ j \in S^c}} (z_j - z_i)}= (-1)^{|\tilde{\lambda}|}\sum_{\sigma\in\mathfrak{S}_n}(-1)^{\sgn(\sigma)}[\prod_{i=1}^{n}z_i^{V_{\sigma(i),i}}]P(z_1,z_2,\dots,z_n)
\end{equation*}
where $[z_1^{c_1}z_2^{c_2}\cdots z_n^{c_n}]P(z_1,z_2,\dots,z_n)$ represents the operation of taking the coefficient and
\begin{align*}
    V_{i,j}=\begin{cases*}
        (\lambda_{n-k+1-j})+j-i \qquad \text{if $j\leq n-k$}\\
        j-i \qquad \text{if $j>n-k$}
    \end{cases*}.
\end{align*}

\end{lem}

Note that we can write \eqref{eq: rec final} as
\begin{equation}\label{eq:final}
    (-1)^{|\tilde{\lambda}|}\stat_{(\delta_S,f^{z}_{S})}(R) \times \sum_{S \in \binom{[n]}{k}} \dfrac{s_{\tilde{\lambda}}[z_S] B_S}{\prod_{\substack{i \in S \\ j \in S^c}} (z_j - z_i)},
\end{equation}
where \(
    B_S=\frac{1}{\stat_{(\Rec^{\leq N},g^{z}_{S})}(w_0)}\sum_{\substack{R\in \Sub(\Rec^{\leq N},|\lambda|)\\ \gamma(R)=0}}\stat_{(\Rec^{\leq N},g^{z}_{S})}(R)x^{R}.\)
To apply \Cref{lem: key connection JT formula} to \eqref{eq:final}, we now describe a monomial coefficient of $B=B_{[n-k+1,n]}$. For $R\in \Sub(\Rec^{\leq N},|\lambda|)$ denoted by $R:D\rightarrow \mathbb{Z}_{\geq1}$, we associate $\eta(R)=(L_1,L_2,\dots,L_n)$ where $L_i$ is a subset of $\mathbb{Z}_{\geq 0}\times \mathbb{Z}_{\geq 1}$ given by: $(c,d)\in L_i$ if and only if $(c+1,i)\in D$ and $R(c+1,i)=d$. 
\begin{lem}\label{lem: lemmas for reduced assignments}
For $R\in \Sub(\Rec^{\leq N},|\lambda|)$ with $\gamma(R)=0$, $\alpha=\rsum(R)$, and $\beta=\csum(R)$,
\begin{enumerate}
    \item denoting $\eta(R)=(L_1,L_2,\dots,L_n)$, we have 
$L_i\in \mathbf{C}_{\beta_i}$ if $i\leq n-k$ and $L_i\in \hat{\mathbf{C}}_{\beta_i}$ if $i> n-k$
    \item we have \(
        \stat_{(\Rec^{\leq N},g^{z}_{S})}(R)=\frac{ \stat_{(\Rec^{\leq N},g^{z}_{S})}(w_0)}{q^{n|\lambda|}\prod_{i=1}^{N-1}z_{N+n-i}^{\alpha_i}} q^{\dinv(\eta(R))}\prod_{i=1}^{n}z_i^{\beta_i}\) for $S=\{n-k+1,n-k+2,\dots,n\}$.
\end{enumerate}
\end{lem}
We let $\Phi$ be an evaluation map given by: $y_{i,j}\rightarrow z_{N+n-i-1}^{-1}x_j$ if $0\leq i\leq N-2$ and $y_{i,j}\rightarrow 0$ if $i>N-2$. By \Cref{lem: lemmas for reduced assignments}, we have
\(
    [\prod_{i=1}^{n}z_i^{\beta_i}]B=\frac{1}{q^{n|\lambda|}}\Phi \left(\prod_{i=1}^{n-k}\mathfrak{h}_{\beta_i}\prod_{i=n-k+1}^{n}\hat{\mathfrak{h}}_{\beta_i} \cdot 1 \right).
\)
By \Cref{lem: key connection JT formula}, \eqref{eq:final} equals to
\(
    \frac{\stat_{(\delta_S,f^{z}_{S})}(R)}{q^{n|\lambda|}}\Phi\left(\det(W) \cdot 1 \right),
\)
where $W=W^{(s)}$ ($k$ is replaced by $n-k$) is defined in \eqref{eq: Wr matrix formula}.

\begin{proof}[Proof of \Cref{thm: main theorem} (c)] In the expansion $\det(W) \cdot 1$, a variable $y_{i,j}$ for $i>N-2$ does not appear as $N$ is sufficiently large. We may only consider the specialization $y_{i,j}\rightarrow z_{N+n-i-1}^{-1}x_j$ and again specializing with \eqref{eq: z_i values}, gives $y_{i,j}\rightarrow q^{n} t^{i}x_j$. Note that $\det(W) \cdot 1$ is of (homogenous) degree $|\lambda|$ in $\mathbf{y}$'s, comparing with \eqref{eq: lw formula final}, we have \(\frac{\stat_{(\delta_S,f^{z}_{S})}(R)}{q^{n|\lambda|}}\Phi\left(\det(W) \cdot 1 \right)\) specialize by \eqref{eq: z_i values} \(\frac{T_{\delta^{[n]}} (q^{n})^{|\lambda|}}{q^{n|\lambda|}}\left(\det(W) \cdot 1 |_{y_{i,j} = t^i x_j}\right)=T_{\delta^{[n]}} (-1)^{\adj(\lambda)}\LW_{\lambda}.\)
Together with \Cref{lem: final rec}, the proof is complete.
\end{proof}

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