\documentclass[ALCO,ThmDefs,Unicode,epreuves]{cedram}
\OneNumberAllTheorems

\usepackage{xcolor}

\usepackage[vcentermath,enableskew]{youngtab}
\usepackage{color}
\usepackage[noadjust]{cite} 
\usepackage{tikz}

%%%
\newenvironment{theorem1*}{\begin{enonce*}{Theorem~\ref{thm:main}}}{\end{enonce*}}
\newenvironment{theorem2*}{\begin{enonce*}{Theorem~\ref{thm:cw}}}{\end{enonce*}}
\newenvironment{theorem3*}{\begin{enonce*}{Theorem~\ref{cor:main}}}{\end{enonce*}}
%%%

%%%
\newcommand*{\C}{\mathbb{C}}
\renewcommand\S{\mathfrak{S}}
\newcommand\Z{\mathbb{Z}}
\newcommand\hd{{\widehat{D}}}
\newcommand\hc{{\widehat{C}}}
\newcommand*{\Schub}{\mathfrak{S}}
%%%

%%%
\DeclareMathOperator*{\Span}{Span}
\DeclareMathOperator*{\charrm}{char}
\DeclareMathOperator*{\tr}{tr}
\DeclareMathOperator*{\GL}{GL}
\DeclareMathOperator*{\purple}{Purple}
\DeclareMathOperator*{\Purple}{\mathbf{P}}
\DeclareMathOperator*{\perm}{perm}
\DeclareMathOperator*{\aug}{aug}
\DeclareMathOperator*{\diag}{diag}
%%%

\newcommand{\arxiv}[1]{\href{http://arxiv.org/abs/#1}{\textsf{arXiv:#1}}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand*{\mk}{\mkern -1mu}
\newcommand*{\Mk}{\mkern -2mu}
\newcommand*{\mK}{\mkern 1mu}
\newcommand*{\MK}{\mkern 2mu}

\hypersetup{urlcolor=purple, linkcolor=blue, citecolor=red}


\newcommand*{\romanenumi}{\renewcommand*{\theenumi}{\roman{enumi}}}
\newcommand*{\Romanenumi}{\renewcommand*{\theenumi}{\Roman{enumi}}}
\newcommand*{\alphenumi}{\renewcommand*{\theenumi}{\alph{enumi}}}
\newcommand*{\Alphenumi}{\renewcommand*{\theenumi}{\Alph{enumi}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% Auteur

%%%1
\author{\firstname{Karola} \lastname{M{\'e}sz{\'a}ros}}

\address{Department of Mathematics\\ 
Cornell University\\ 
Ithaca, NY 14853\\
USA.} 

\email{karola@math.cornell.edu}

%%%2
\author{\firstname{Arthur} \lastname{Tanjaya}}

\address{Department of Mathematics\\ 
Cornell University\\ 
Ithaca NY 14853\\
USA.} 

\email{amt333@cornell.edu}

\thanks{Karola M{\'e}sz{\'a}ros is partially supported by CAREER NSF Grant DMS-1847284.}

%%%%% Sujet

\keywords{Schubert polynomial, principal specialization, nonnegative linear combination.}

\subjclass{05E05}

%%%%% Gestion

\DOI{10.5802/alco.200}
\datereceived{2021-05-06}
\daterevised{2021-10-21}
\dateaccepted{2021-10-27}


%%%%% Titre et résumé
\title
[Inclusion-exclusion on Schubert polynomials]
{Inclusion-exclusion on Schubert polynomials}

\begin{abstract}
We prove that an inclusion-exclusion inspired expression of Schubert polynomials of permutations that avoid the patterns $1432$ and $1423$ is nonnegative. Our theorem implies a partial affirmative answer to a recent conjecture of Yibo Gao about principal specializations of Schubert polynomials. We propose a general framework for finding inclusion-exclusion inspired expression of Schubert polynomials of all permutations. 
\end{abstract}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{document}

\maketitle

\section{Introduction}
Schubert polynomials, introduced by Lascoux and Sch{\"u}tzenberger in~\cite{LS1}, represent cohomology classes of Schubert cycles in the flag variety. They are also multidegrees of matrix Schubert varieties~\cite{multidegree} and wield an impressive collection of combinatorial formulas~\cite{laddermoves, BJS, FK1993, nilcoxeter, thomas, lenart, manivel, prismtableaux}. Yet, only recently have their supports been established as integer points of generalized permutahedra~\cite{FMS, MTY}. There have also been several exciting recent developments about the coefficients of Schubert polynomials: 
\begin{enumerate} 
\item % (1) 
they are known to be log-concave along root directions in their Newton polytopes~\cite{june}; 
\item % (2) 
the set of permutations whose Schubert polynomials have all their coefficients less than or equal to a fixed integer $m$ is closed under pattern containment~\cite{zeroone}. 
\end{enumerate}
Recall that $\pi=\pi_1\ldots \pi_k \in S_k$ is a pattern of $\sigma=\sigma_1\ldots \sigma_n \in S_n$ if and only if there are indices $1\leq i_1<i_2<\cdots<i_k\leq n$ so that the relative order of 
$\pi_1, \ldots, \pi_k$ and of $\sigma_{i_1}, \ldots, \sigma_{i_k}$ are the same. 

\subsection{Nonnegative linear combinations of Schubert polynomials with monomial coefficients} 

\looseness-1 
In this paper we investigate nonnegativity properties of linear combinations of Schubert polynomials with monomial coefficients in $\Z[x_1, \ldots, x_n]$ associated to patterns of a fixed permutation. A first step in this direction is a recent result by Fink, St. Dizier and the first author of the present paper: 

\begin{theorem}[{\cite[Theorem~1.2]{zeroone}}] \label{thm:fms} 
Fix $\sigma \in S_n$ and let $\pi \in S_{n-1}$ be the pattern of $\sigma$ with Rothe diagram $D(\pi)$ obtained by removing row $k$ and column $\sigma_k$ from $D(\sigma)$. Then
\begin{align} \label{eq:?}
\mathfrak{S}_{\sigma}(x_1, \ldots, x_n)-M_{\sigma, \pi}(x_1, \ldots, x_n) \mathfrak{S}_{\pi}(x_{1}, \ldots, \widehat{x_k}, \ldots, x_{n}) \in \mathbb{Z}_{\geq 0}[x_1, \ldots, x_n] 
\end{align} 
where 
\[
M_{\sigma, \pi}(x_1, \ldots, x_n) = \left(\prod_{(k,i)\in D(\sigma)}{x_k}\right)\left(\prod_{(i,\sigma_k)\in D(\sigma)}{x_i} \right).
\]
\end{theorem}

In particular, Theorem~\ref{thm:fms} implies that the set of permutations whose Schubert polynomials have all their coefficients less than or equal to a fixed integer $m$ is closed under pattern containment.

%\medskip

The first result of this paper is a broad extension of Theorem~\ref{thm:fms} for $1432$ and $1423$ avoiding permutations: 

\begin{theorem} \label{thm:main}
Let $w \in S_n$ be a $1432$ and $1423$ avoiding permutation and let $u$ be a subword of $w$. Then
\begin{equation} \label{eqn:f_w1}
\sum_{u \le v \le w} (-1)^{\left|w\right|-\left|v\right|} M_{w, v} \Schub_{\perm(v)}(\mathbf{x}_{w^{-1}(v)}) \in \Z_{\ge0}[x_1, \dots, x_n],
\end{equation}
where
\begin{equation*}
M_{w, v} \coloneqq \prod_{(i, j) \in D(w) \setminus \widehat{D(w)}\mid_{v}} x_i.
\end{equation*}
\end{theorem}

In Theorem~\ref{thm:main} we use the relation of containment on words: for words $u, v$, we say $u \le v$ if $u$ occurs as a subword in $v$. Moreover, for a word $v$ of length $n$, $\pi = \perm(v)$ is the permutation in $S_n$ such that the relative order of $\pi_1, \dots, \pi_n$ and of $v_1, \dots, v_n$ are the same.
For these and other definitions used in Theorems~\ref{thm:fms} and~\ref{thm:main} see Sections~\ref{sec:background} and~\ref{sec:main} which lay them out in detail. Here we give an example of Theorem~\ref{thm:main} for illustration.
For $w = 1342$ and $u = 42$ we have
$\{ v \mid u \le v \le w \} = \{ 1342, 142, 342, 42 \},$
so the alternating sum in~\eqref{eqn:f_w1} becomes
\begin{multline*}
M_{w, 1342}\Schub_{1342}(x_1,x_2, x_3, x_4)\\ - M_{w, 142}\Schub_{132}(x_1, x_3, x_4)
- M_{w, 342}\Schub_{231}(x_2, x_3, x_4) + M_{w, 42}\Schub_{21}(x_3, x_4) \\
\begin{aligned}
&= 1 \cdot (x_1x_2 + x_1x_3 + x_2x_3) - x_2 \cdot (x_1 + x_3) - 1 \cdot (x_2 x_3) + x_2 \cdot (x_3) \\
&= x_1x_3,
\end{aligned}
\end{multline*}
which indeed has nonnegative coefficients. See Figure~\ref{fig2} for an illustration. 
%\medskip

An immediate corollary of Theorem~\ref{thm:main} is the following theorem: 
\begin{theorem} \label{cor:main}
Let $w \in S_n$ be a $1432$ and $1423$ avoiding permutation. If $u$ is a subword of $w$, then
\begin{equation} \label{eqn:cw}
\sum_{u \le v \le w} (-1)^{\left|w\right|-\left|v\right|} \Schub_{\perm (v)}(\mathbf{1})\geq 0,
\end{equation}
where $\Schub_{\perm (v)}(\mathbf{1})$ denotes the value of the Schubert polynomial $\Schub_{\perm (v)}$ with all its variables set to $1$.
\end{theorem}

Theorem~\ref{cor:main} is closely related to a recent conjecture of Gao~\cite[Conjecture~3.2]{Gao} regarding the principal specialization of Schubert polynomials as we now explain. We also conjecture (Conjecture~\ref{conj:1.3}) in Section~\ref{sec:conj} that Theorem~\ref{cor:main} holds for all permutations $w \in S_n$. 

\subsection{Principal specializations of Schubert polynomials} Macdonald~\cite[Eq.~6.11]{macdonald} famously expressed the principal specialization $\Schub_{\sigma}(\bf{1})$ of the Schubert polynomial $\Schub_{\sigma}$ in terms of the reduced words of $\sigma$. Fomin and Kirillov~\cite{reduced} placed this expression in the context of plane partitions for dominant permutations, while after two decades Billey et al.~\cite{bijmac} provided a combinatorial proof. In 2017, Stanley~\cite{she} considered the asymptotics of $\Schub_{\sigma}(\bf{1})$ as well as the role pattern containment plays in its value. The asymptotics question was partially answered by Morales, Pak and Panova~\cite{pak}, while the pattern avoidance question inspired Weigandt~\cite{132} and Gao~\cite{Gao}, among others, to seek an understanding of $\Schub_{\sigma}(\bf{1})$ in terms of the permutation patterns of $\sigma$. Weigandt showed that $\Schub_{\sigma}(\mathbf{1})\geq 1+p_{132}(\sigma)$, where $p_{\pi}(\sigma)$ is the number of patterns $\pi$ in the permutation $\sigma$, while Gao improved this to $\Schub_{\sigma}(\mathbf{1})\geq 1+p_{132}(\sigma)+p_{1432}(\sigma)$. Gao conjectured that there exist nonnegative integers $c_w$, for $w \in S_{\infty}$, such that 
\[
\Schub_{\sigma}(\mathbf{1})=\sum_{\pi \in S_{\infty}} c_{\pi}p_{\pi}(\sigma).
\] 

Equivalently:

\begin{conjecture}[{\cite[Conjecture~3.2]{Gao}}])\label{conj:gao} 
There exist nonnegative integers $c_w$, for $w \in S_{\infty}$, such that 
\[
\Schub_w(\mathbf{1})=\sum_{v \leq w} c_{\perm(v)},
\]
where 
$v \leq w$ denotes that $v$ occurs as a subword in $w$. 
\end{conjecture}

It follows readily via inclusion-exclusion that 
for $w \in S_{\infty}$: 
\begin{equation} \label{eq:cw} 
c_w=\sum_{v \le w} (-1)^{\left|w\right|-\left|v\right|} \Schub_{\perm(v)}(\bf{1}).
\end{equation} 
Thus, Theorem~\ref{cor:main} settles Gao's conjecture~\ref{conj:gao} for $1432$ and $1423$ avoiding permutations $w \in S_{\infty}$ when we specialize it to the empty word $u=()$. Moreover, we also 
provide a combinatorial interpretation of the numbers $c_w$ for $1432$ and $1423$ avoiding permutations $w \in S_{\infty}$:

\begin{theorem} \label{thm:cw} 
For $1432$ and $1423$ avoiding permutations $w \in S_{\infty}$ the value of $c_w$ is the number of diagrams $C \le D(w)$ that cannot be written as $\widehat{C}_{\aug}$ for some $\widehat{C}=\widehat{C}_{k, w_k} \le \widehat{D(w)}=\widehat{D(w)}_{k, w_k}$, $k \in \mathbb{Z}_{>0}$. 
\end{theorem}

See Section~\ref{sec:gao} for more details.


\subsection{Extending Theorems~\ref{thm:fms} \&~\ref{thm:main}} Both Theorem~\ref{cor:main} and Theorem~\ref{thm:cw} are byproducts of our main Theorem~\ref{thm:main}. It is thus most natural to ask in what generality Theorem~\ref{thm:main} holds. While Theorem~\ref{cor:main} is conjectured by Gao to hold for all permutations, Theorems~\ref{thm:main} and~\ref{thm:cw} as stated do not. Theorem~\ref{thm:main} fails already for $w=1432$. However, the reason it fails leads to other possibilities: the monomials $M_{w,v}$ we used to formulate Theorem~\ref{thm:main} are inspired by Theorem~\ref{thm:fms} and are one of many choices we might have made. While Fink, M\'esz\'aros, and St.~Dizier~\cite{zeroone} only constructed one monomial $M_{\sigma, \pi}$ for the pair of permutations $(\sigma, \pi)$ in Theorem~\ref{thm:fms}, there is a family of monomials each of which would make~\eqref{eq:?} true. We are led to wonder whether for an appropriate choice of such monomials Theorem~\ref{thm:main} could be generalized to any permutation. We take the first step towards this goal via the following generalization of Theorem~\ref{thm:fms} showing that a family of monomials, including $M_{\sigma, \pi}$ could work:

\begin{theorem} \label{thm:gen}
Fix $\sigma \in S_n$ and let $\pi \in S_{n-1}$ be the pattern of $\sigma$ with Rothe diagram $D(\pi)$ obtained by removing row $k$ and column $\sigma_k$ from $D(\sigma)$. 
If $K \in \Purple_{k, \sigma_k}(D(\sigma))$, then
\[
\mathfrak{S}_{\sigma}(x_1, \ldots, x_n)-M(x_1, \ldots, x_n) \mathfrak{S}_{\pi}(x_{1}, \ldots, \widehat{x_k}, \ldots, x_{n}) \in \mathbb{Z}_{\geq 0}[x_1, \ldots, x_n],
\] 
where 
\[
M(x_1, \dots, x_n) = \prod_{(i, j) \in K} x_i.
\]
\end{theorem}

See Section~\ref{sec:gen} for the definition of the set of diagrams $\Purple_{k, \sigma_k}(D(\sigma))$ used in the statement of Theorem~\ref{thm:gen} above and Section~\ref{sec:conj} for a discussion of how Theorem~\ref{thm:gen} could be used to generalize Theorem~\ref{thm:main} as well as Conjecture~\ref{conj:16} examining the strength of Theorem~\ref{thm:gen}. 

\subsection*{Outline of this paper}
Section~\ref{sec:background} lays out the general background on Schubert polynomials that we rely on. Section~\ref{sec:main} contains the setup and proofs of Theorem~\ref{thm:main}, \ref{cor:main} and~\ref{thm:cw}. Section~\ref{sec:gen} provides a proof of Theorem~\ref{thm:gen} and its generalization Theorem~\ref{thm:gen1}, while Section~\ref{sec:conj} concludes with conjectures and open problems.

\section{Background on Schubert polynomials}
\label{sec:background}

Schubert polynomials were originally defined via divided difference operators. We will instead define them as dual characters of flagged Weyl modules for Rothe diagrams. This section follows the exposition of~\cite{FMS, zeroone}. 

\subsection{Definition of dual characters of flagged Weyl modules}
A \emph{diagram} is a sequence $D = (C_1, C_2, \ldots,$ $C_n)$ of finite subsets of $[n]$, called the \emph{columns} of $D$. We interchangeably think of $D$ as a collection of boxes $(i,j)$ in a grid, viewing an element $i\in C_j$ as a box in row $i$ and column $j$ of the grid. When we draw diagrams, we read the indices as in a matrix: $i$ increases top-to-bottom and $j$ increases left-to-right. 

The \emph{Rothe diagram} $D(w)$ of a permutation $w\in S_n$ is the diagram
\[
D(w)=\{(i,j)\in [n]\times [n] \mid i<(w^{-1})_j\mbox{ and } j<w_i \}.
\]
Note that Rothe diagrams have the \emph{northwest property}: If $(r,c'),(r',c)\in D(w)$ with $r<r'$ and $c<c'$, then $(r,c)\in D(w)$.

Let $G=\GL(n,\mathbb{C})$ be the group of $n\times n$ invertible matrices over $\mathbb{C}$ and $B$ be the subgroup of $G$ consisting of the $n\times n$ upper-triangular matrices. The flagged Weyl module is a representation $\mathcal{M}_D$ of $B$ associated to a diagram $D$. The dual character of $\mathcal{M}_D$ has been shown in certain cases to be a Schubert polynomial~\cite{KP} or a key polynomial~\cite{flaggedLRrule}. We will use the construction of $\mathcal{M}_D$ in terms of determinants given in~\cite{magyar}.

Denote by $Y$ the $n\times n$ matrix with indeterminates $y_{ij}$ in the upper-triangular positions $i\leq j$ and zeros elsewhere. Let $\mathbb{C}[Y]$ be the polynomial ring in the indeterminates $\{y_{ij}\}_{i\leq j}$. Note that $B$ acts on $\mathbb{C}[Y]$ on the right via left translation: if $f(Y)\in \mathbb{C}[Y]$, then a matrix $b\in B$ acts on $f$ by $f(Y)\cdot b=f(b^{-1}Y)$. For any $R,S\subseteq [n]$, let $Y_S^R$ be the submatrix of $Y$ obtained by restricting to rows $R$ and columns $S$.

For $R,S\subseteq [n]$, we say $R\leq S$ if $\#R=\#S$ and the $k$\/th least element of $R$ does not exceed the $k$\/th least element of $S$ for each $k$. For any diagrams $C=(C_1,\ldots, C_n)$ and $D=(D_1,\ldots, D_n)$, we say $C\leq D$ if $C_j\leq D_j$ for all $j\in[n]$.

\begin{defi}
For a diagram $D=(D_1,\ldots, D_n)$, the \emph{flagged Weyl module} $\mathcal{M}_D$ is defined by
\[
\postdisplaypenalty1000000
\mathcal{M}_D=\Span\nolimits_\mathbb{C}\left\{\prod_{j=1}^{n}\det\left(Y_{D_j}^{C_j}\right)\ \middle|\ C\leq D \right\}.
\]
$\mathcal{M}_D$ is a $B$-module with the action inherited from the action of $B$ on $\mathbb{C}[Y]$. 
\end{defi}

Note that since $Y$ is upper-triangular, the condition $C\leq D$ is technically unnecessary since $\det\left(Y_{D_j}^{C_j}\right)=0$ unless $C_j\leq D_j$. Conversely, if $C_j\leq D_j$, then $\det\left(Y_{D_j}^{C_j}\right)\neq 0$. 

For any $B$-module $N$, the \emph{character} of $N$ is defined by $\charrm(N)(x_1,\ldots,x_n)=\tr \left(X:N\to N\right)$, where $X$ is the diagonal matrix $\diag(x_1,x_2,\ldots,x_n)$ with diagonal entries $x_1,\ldots,x_n$, and $X$ is viewed as a linear map from $N$ to $N$ via the $B$-action. Define the \emph{dual character} of $N$ to be the character of the dual module $N^*$:
\begin{align*}
\charrm\nolimits^*(N)(x_1,\ldots,x_n)&=\tr \left(X:N^*\to N^*\right) \\
&=\charrm(N)(x_1^{-1},\ldots,x_n^{-1}).
\end{align*}

\begin{defi}
For a diagram $D\subseteq [n]\times [n]$, let $\chi_D=\chi_D(x_1,\ldots,x_n)$ be the dual character 
\[
\chi_D=\charrm\nolimits^*\mathcal{M}_D. 
\] 
\end{defi}

\subsection{Results about dual characters of flagged Weyl modules}

A special case of dual characters of flagged Weyl modules of diagrams are Schubert polynomials:

\begin{theorem}[\cite{KP}]\label{thm:kp}
For $w$ a permutation and $D(w)$ its Rothe diagram we have that the Schubert polynomial $\S_w$ is 
\[
\mathfrak{S}_w = \chi_{D(w)}. 
\]
\end{theorem}

\begin{theorem}[\cf {\cite[Theorem~7]{FMS}}]
For any diagram $D\subseteq [n]\times [n]$, the monomials appearing in $\chi_D$ are exactly 
\[
\left\{\prod_{j=1}^{n}\prod_{i\in C_j}x_i\ \middle|\ C\leq D \right\}.
\]
\end{theorem}

\begin{theorem}[\cite{zeroone}]\label{cor:fms}
\looseness-1
Let $D\subseteq [n]\times [n]$ be a diagram. Fix any diagram $C^{(1)}\leq D$ and set 
\[
\mathbf{m}=\prod_{j=1}^{n}\prod_{i\in C^{(1)}_j}x_i.
\] 
Let $C^{(1)}, \ldots, C^{(r)}$ be all the diagrams $C$ such that $C\leq D$ and $\prod_{j=1}^{n}\prod_{i\in C_j}x_i=\mathbf{m}$. Then, the coefficient of $\mathbf{m}$ in $\chi_D$ is equal to 
\[
[\mathbf{m}]\chi_D =\dim \left(\Span\nolimits_\mathbb{C}\left\{\prod_{j=1}^{n}\det\left(Y_{D_j}^{C^{(i)}_j}\right) \ \middle|\ i\in [r] \right\}\right).
\] 
In particular, 
\[
[\mathbf{m}]\chi_D \leq \#\left\{ C \le D \;\middle|\; \prod_{(i, j) \in C} x_i = \mathbf{m} \right\}.
\] 
\end{theorem}

\goodbreak
In light of the last inequality, it is natural to wonder when equality holds. This is what Fan \& Guo~\cite{FG} did: 


\begin{theorem}[\cite{FG}]\label{thm:fg}
Given a diagram $D \subseteq [n] \times [n]$, let
\[
x^D = \prod_{(i, j) \in D} x_i.
\]
Then, for a permutation $w \in S_n$,
\[
\Schub_w(x_1, \dots, x_n) = \sum_{C \le D(w)} x^C
\]
if and only if $w$ avoids the patterns $1432$ and $1423$.
\end{theorem}

In particular, Theorem~\ref{thm:fg} implies:

\begin{coro}[\cite{FG}]\label{cor:counting}
If $w \in S_n$ avoids the patterns $1432$ and $1423$, then the coefficient of $\mathbf{m}$ in $\Schub_w = \chi_{D(w)}$ is equal to
\[
[\mathbf{m}]\Schub_w=\# \left\{ C \le D(w) \;\middle|\; \prod_{(i, j) \in C} x_i = \mathbf{m} \right\}.
\]
\end{coro}

\section{Proof of Theorems~\ref{thm:main}, \ref{cor:main} and~\ref{thm:cw}}
\label{sec:main}

In this section we prove Theorems~\ref{thm:main}, \ref{cor:main} and~\ref{thm:cw}. We start by giving the necessary definitions and lemmas. 

\subsection{Setup for Theorems~\ref{thm:main}, \ref{cor:main} and~\ref{thm:cw}}
\begin{defi}
For words $u, v$, we write $u \le v$ if $u$ is a subword of $v$ (and $u < v$ if $u \le v$ and $u \ne v$). In other words, $u \le v$ if there is a sequence $1 \le i_1 < \cdots < i_{\left|u\right|} \le \left|v\right|$ such that $u = v(i_1) \cdots v(i_{\left|u\right|})$. The empty word $()$ is a pattern in all words.
\end{defi}

\begin{exam}
Let $u = 792$ and $v = 37952$. Then $u \le v$, because $u = v(2)\,v(3)\,v(5)$.
\end{exam}

\begin{defi}
For a word $v$ of length $n$, let $i_1, i_2, \dots, i_n$ be indices such that $v(i_1) < v(i_2) < \cdots < v(i_n)$. Then $\perm(v)$ is the permutation that sends $i_j \mapsto j$, that is, $\perm(v) = (i_1i_2 \cdots i_n)^{-1}$. Equivalently, $\pi = \perm(v)$ is the permutation in $S_n$ such that the relative order of $\pi_1, \dots, \pi_n$ and of $v_1, \dots, v_n$ are the same.
\end{defi}

\begin{exam}
Let $v = 37952$. Note $v(5) < v(1) < v(4) < v(2) < v(3)$, so $(i_1, i_2, i_3, i_4, i_5) = (5, 1, 4, 2, 3)$. Thus $\perm(v) = (51423)^{-1} = 24531$. Notice that we can obtain $\perm(v)$ from $v$ by replacing the smallest character of $v$ with $1$, the second smallest with $2$, and so on.
\end{exam}

\begin{defi}
Let $w \in S_n$ and let $v$ be a subword of $w$. We define
\[
\mathbf{x}_{w^{-1}(v)} \coloneqq (x_{w^{-1}(v(1))}, x_{w^{-1}(v(2))}, \dots, x_{w^{-1}(v(\left|v\right|))}).
\]
\end{defi}

\begin{exam}
Let $w = 134265$ and $v = 3265$. Then
\[
\mathbf{x}_{w^{-1}(v)} = (x_{w^{-1}(3)}, x_{w^{-1}(2)}, x_{w^{-1}(6)}, x_{w^{-1}(5)}) = (x_2, x_4, x_5, x_6).
\]
Notice that the resulting indices will always be in ascending order. See Figure~\ref{yellow} for an illustration. 
\end{exam}

\begin{figure}[htb] \centering
\includegraphics[width=0.6\textwidth]{Figures/fig_134265_3265.pdf}
\caption{The left diagram is the Rothe diagram of the permutation $w = 134265$ (the permutation $w$ is noted in red to the left of the diagram). The row indices are noted in blue to the left of the diagram. The right diagram shows the subword $v = 3265$ of $w = 134265$ graphically: it is obtained by removing the yellow highlighted rows and columns from the Rothe diagram of $w$. The indices $w^{-1}(v)$ shown in blue to the left of the diagram are simply the row indices corresponding to this graphical presentation of the subword $v = 3265$ of $w = 134265$.}
\label{yellow}
\end{figure}

\goodbreak
\begin{defi}
Given a diagram $D \subseteq [n] \times [n]$ and sets of indices $K, L \subseteq [n]$ with $\#K = \#L$, let $\widehat{D}\mid_{K, L}$ denote the diagram obtained from $D$ by keeping only the boxes in rows $K$ and columns $L$:
\[
\widehat{D}\mid_{K, L} = \{ (i, j) \in D \mid i \in K, j \in L \}.
\]
\end{defi}

\begin{defi}
Suppose $C \le D(w)$ for some permutation $w \in S_n$. Then, for any subword $v \le w$, we define $\widehat{C}\mid_{v}$ to be the diagram obtained by keeping only the boxes in the rows corresponding to $v$. That is, $\widehat{C}\mid_{v} \coloneqq \widehat{C}\mid_{K, L}$, where $L = \{v(1), v(2), \dots, v(\left|v\right|)\}$ and $K = \{w^{-1}(v(1)), w^{-1}(v(2)), \dots, w^{-1}(v(\left|v\right|))\}$.
\end{defi}

\subsection{Theorem~\ref{thm:main} and its proof}

\begin{theorem1*} 
Let $w \in S_n$ be a $1432$ and $1423$ avoiding permutation and let $u$ be a subword of $w$. Then
\begin{equation} \label{eqn:f_w}
\sum_{u \le v \le w} (-1)^{\left|w\right|-\left|v\right|} M_{w, v} \Schub_{\perm(v)}(\mathbf{x}_{w^{-1}(v)}) \in \Z_{\ge0}[x_1, \dots, x_n],
\end{equation}
where
\begin{equation*}
M_{w, v} \coloneqq \prod_{(i, j) \in D(w) \setminus \widehat{D(w)}\mid_{v}} x_i.
\end{equation*}
\end{theorem1*}

\begin{exam} \label{ex1}
Let $w = 2143$ and $u = 43$. Then
\[
\{ v \mid u \le v \le w \} = \{ 2143, 143, 243, 43 \},
\]
so the alternating sum in~\eqref{eqn:f_w} becomes 
\begin{multline} \label{?1}
M_{w, 2143}\Schub_{2143}(x_1, x_2, x_3, x_4) 
\\
- M_{w, 143}\Schub_{132}(x_2, x_3, x_4) 
- M_{w, 243}\Schub_{132}(x_1, x_3, x_4) + M_{w, 43}\Schub_{21}(x_3, x_4) \\ 
\begin{aligned}
& = 1 \cdot (x_1^2 + x_1x_2 + x_1x_3) - x_1 \cdot (x_2 + x_3) - x_1 \cdot (x_1 + x_3) + x_1 \cdot (x_3) \\
&= 0,
\end{aligned}
\end{multline}
which indeed has nonnegative coefficients. See Figure~\ref{fig1} for an illustration.
\end{exam}

\begin{figure}[htb]\centering
\includegraphics[width=\textwidth]{Figures/fig_2143_43.pdf}
\caption{The four diagrams in this figure correspond left to right to the subwords $\{ v \mid u \le v \le w \} = \{ 2143, 143, 243, 43 \}$ for $w = 2143$ and $u = 43$ as in Example~\ref{ex1}. 
These in turn yield the Schubert polynomials in the expression~\eqref{?1}. The red numbers on the left of the diagrams signify these subwords; the blue numbers are the row numbers yielding the variables of the corresponding Schubert polynomials in the expression~\eqref{?1}. The purple boxes correspond to the boxes of the Rothe diagram of $w = 2143$ that are removed in order to obtain $v$; graphically these are the boxes struck by yellow if the yellow highlighted rows and columns are extended; the row indices of these boxes yield the monomials $M_{w,v}$.}
\label{fig1}
\end{figure}

\goodbreak
\begin{exam} \label{ex2}
Let $w = 1342$ and $u = 42$. Then
\[
\{ v \mid u \le v \le w \} = \{ 1342, 142, 342, 42 \},
\]
so the alternating sum in~\eqref{eqn:f_w} becomes 
\begin{multline} \label{?2}
M_{w, 1342}\Schub_{1342}(x_1,x_2, x_3, x_4) \\
- M_{w, 142}\Schub_{132}(x_1, x_3, x_4) 
- M_{w, 342}\Schub_{231}(x_2, x_3, x_4) + M_{w, 42}\Schub_{21}(x_3, x_4) \\ 
\begin{aligned} 
&= 1 \cdot (x_1x_2 + x_1x_3 + x_2x_3) - x_2 \cdot (x_1 + x_3) - 1 \cdot (x_2 x_3) + x_2 \cdot (x_3) \\ 
&= x_1x_3,
\end{aligned}
\end{multline} 
which indeed has nonnegative coefficients. See Figure~\ref{fig2} for an illustration.
\end{exam}

\begin{figure}[htb]\centering
\includegraphics[width=\textwidth]{Figures/fig_1342_42.pdf}
\caption{The four diagrams in this figure correspond left to right to the subwords $\{ v \mid u \le v \le w \} = \{ 1342, 142, 342, 42 \}$ for $w = 1342$ and $u = 42$ as in Example~\ref{ex2}. 
These in turn yield the Schubert polynomials in the expression~\eqref{?2}. The red numbers on the left of the diagrams signify these subwords; the blue numbers are the row numbers yielding the variables of the corresponding Schubert polynomials in the expression~\eqref{?2}. The purple boxes correspond to the boxes of the Rothe diagram of $w = 1342$ that are removed in order to obtain $v$; graphically these are the boxes struck by yellow if the yellow highlighted rows and columns are extended; the row indices of these boxes yield the monomials $M_{w,v}$.}
\label{fig2}
\end{figure}

To aid the proof of Theorem~\ref{thm:main} we extend Corollary~\ref{cor:counting} to words:

\begin{lemma}
\label{lem:counting}
Let $w \in S_n$ be a $1432$ and $1423$ avoiding permutation, and let $v$ be a subword of $w$. Then the coefficient of $\mathbf{m}$ in $\Schub_{\perm(v)}(\mathbf{x}_{w^{-1}(v)})$ is equal to
\[
\# \left\{ C \le \widehat{D(w)}\mid_{v} \;\middle|\; \prod_{(i, j) \in C} x_i = \mathbf{m} \text{ and $C$'s boxes all lie in rows $K$} \right\},
\]
where $K = \{ w^{-1}(v(1)), w^{-1}(v(2)), \dots, w^{-1}(v(\left|v\right|)) \}$.
\end{lemma}
\begin{proof}
Fix $\mathbf{m}$, and let
\[
A = \left\{ C \le \widehat{D(w)}\mid_{v} \;\middle|\; \prod_{(i, j) \in C} x_i = \mathbf{m} \text{ and $C$'s boxes all lie in rows $K$} \right\}.
\]
If $\mathbf{m}$ is divisible by some $x_i$ where $i \notin K$, then the coefficient of $\mathbf{m}$ in $\Schub_{\perm(v)}(\mathbf{x}_{w^{-1}(v)})$ is $0$, and no diagram $C$ with boxes only in rows $K$ can ever satisfy $\prod_{(i, j) \in C} x_i = \mathbf{m}$, so $\left|A\right| = 0$ and we are done.

Let $k_1 < k_2 < \cdots < k_{\left|v\right|}$ be the elements of $K$. By the previous discussion, we may as well assume that we can write
\[
\mathbf{m} = \prod_{i = 1}^{\left|v\right|} x_{k_i}^{\alpha_i}
\]
for some nonnegative integers $\alpha_i$. Define
\[
\mathbf{m}' = \prod_{i = 1}^{\left|v\right|} x_i^{\alpha_i},
\]
which is simply $\mathbf{m}$ under the reindexing $x_{k_i} \mapsto x_i$. Since $w$ is 1432 and 1423 avoiding, so are $v$ and $\perm(v)$, thus by Corollary~\ref{cor:counting}, the coefficient of $\mathbf{m'}$ in $\Schub_{\perm(v)}(x_1, \dots, x_{\left|v\right|})$ is equal to $\left|B\right|$, where
\[
B = \left\{ C \le D(\perm(v)) \;\middle|\; \prod_{(i, j) \in C} x_i = \mathbf{m'} \right\}.
\]

Consider the function $f \colon B \to A$ given by
\[
f(C) = \{ (k_i, v(j)) \mid (i, j) \in C \}.
\]
Notice that the boxes of $f(C)$ all lie in rows $K$, and since $\prod_{(i, j) \in C} x_i = \mathbf{m'}$, we have $\prod_{(i, j) \in f(C)} x_i = \mathbf{m}$. Furthermore, from the definition of Rothe diagrams, if $(i, j) \in D(\perm(v))$ then $(k_i, v(j)) \in D(w)$, so $f(D(\perm(v))) \le \widehat{D(w)}\mid_{v}$. For $C, C' \in B$, observe that if $C \le C'$ then $f(C) \le f(C')$, so it follows that $f(C) \le \widehat{D(w)}\mid_{v}$ for all $C \in B$ and $f$ is well-defined.

$f$ is clearly injective by construction. To see that it is surjective, note that if $C \in A$, then $C \le \widehat{D(w)}\mid_{v}$, so every box in $C$ is of the form $(k_i, v(j))$ and the diagram
\[
C' = \{ (i, j) \mid (k_i, v(j)) \in C \}
\]
is easily seen to be a member of $A$, with $f(C') = C$. Therefore,
\begin{align*}
[\mathbf{m}] \Schub_{\perm(v)}(\mathbf{x}_{w^{-1}(v)}) &= [\mathbf{m}] \Schub_{\perm(v)}(x_{k_1}, x_{k_2}, \dots, x_{k_{\left|v\right|}}) \\
&= [\mathbf{m'}] \Schub_{\perm(v)}(x_1, x_2, \dots, x_{\left|v\right|}) \\
&= \left|B\right| \\
&= \left|A\right|.\qedhere
\end{align*}
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
We must show that for every monomial $\mathbf{m}$,
\[
[\mathbf{m}]\sum_{u \le v \le w} (-1)^{\left|w\right|-\left|v\right|} M_{w, v} \Schub_{\perm(v)}(\mathbf{x}_{w^{-1}(v)}) \ge 0;
\]
equivalently,
\begin{equation}
\sum_{u \le v \le w} (-1)^{\left|w\right|-\left|v\right|} [\mathbf{m}] M_{w, v} \Schub_{\perm(v)}(\mathbf{x}_{w^{-1}(v)}) \ge 0.
\end{equation}

Fix $\mathbf{m}$. For any subword $v \le w$, let $K_v \coloneqq \{w^{-1}(v(1)), w^{-1}(v(2)), \dots, w^{-1}(v(\left|v\right|))\}$ (the ``rows corresponding to $v$''). Using Lemma~\ref{lem:counting}, we find that
\begin{align*}
&\hspace*{-5mm}[\mathbf{m}] M_{w, v} \Schub_{\perm(v)}(\mathbf{x}_{w^{-1}(v)})\\
&= \left[\frac{\mathbf{m}}{M_{w, v}}\right] \Schub_{\perm(v)}(\mathbf{x}_{w^{-1}(v)}) \\
&= \# \left\{ C \le \widehat{D(w)}\mid_{v} \;\middle|\; \prod_{(i, j) \in C} x_i = \frac{\mathbf{m}}{M_{w, v}} \text{ and $C$'s boxes all lie in rows $K_v$} \right\}.\qedhere
\end{align*}


Consider the two families of sets
\begin{multline*}
\begin{aligned}
A_v &\coloneqq \left\{ C \le \widehat{D(w)}\mid_{v} \;\middle|\; \prod_{(i, j) \in C} x_i = \frac{\mathbf{m}}{M_{w, v}} \text{ and $C$'s boxes all lie in rows $K_v$} \right\}, \\
B_v &\coloneqq \left\{ C \le D(w) \;\middle|\; \prod_{(i, j) \in C} x_i = \mathbf{m}, C \setminus (\widehat{C}\mid_{v}) = D(w) \setminus (\widehat{D(w)}\mid_{v})\right.
\end{aligned}\\
\left.\vphantom{\left|\prod_{(i, j)}\right.}\text{ and $C$'s boxes all lie in rows $K_w$} \right\}.
\end{multline*}

Since $v \le w$, $K_v \subseteq K_w$, and also the boxes of $D(w) \setminus (\widehat{D(w)}\mid_{v})$ all lie in rows $K_w \setminus K_v$. Thus, $D(w) \setminus (\widehat{D(w)}\mid_{v})$ is disjoint from every $C \in A_v$, and so there is an obvious injection $f$ from $A_v$ to $B_v$ defined by
\[
f(C) \coloneqq C \sqcup (D(w) \setminus (\widehat{D(w)}\mid_{v})).
\]
We claim $f$ is surjective. Indeed, given $C \in B_v$, the diagram $C' = C \setminus (D(w) \setminus (\widehat{D(w)}\mid_{v}\nobreak ))$ is easily seen to be a member of $A_v$, and of course $f(C') = C$.

Therefore,
\begin{equation}
[\mathbf{m}] M_{w, v} \Schub_{\perm(v)}(\mathbf{x}_{w^{-1}(v)}) = \left|A_v\right| = \left|B_v\right|,
\end{equation}
and so it suffices to show that
\begin{equation} \label{eqn:wts}
\sum_{u \le v \le w} (-1)^{\left|w\right|-\left|v\right|} \left|B_v\right| \ge 0.
\end{equation}

Notice that, if $u \le v \le v' \le w$, then $B_v \subseteq B_{v'}$, and for all $u \le v, v' \le w$, $B_u \cap B_v = B_{u \wedge v}$, where $u \wedge v$ denotes the maximal word contained in both $u$ and $v$. Let $I = \{ v \mid u \le v \le w \text{ and } \left|v\right| = \left|w\right| - 1 \}$. Then, using inclusion-exclusion, we find that
\begin{align}
\sum_{u \le v \le w} (-1)^{\left|w\right|-\left|v\right|} \left| B_v \right| &= \left|B_w\right| - \sum_{v_1 \in I} \left| B_{v_1} \right| + \sum_{v_1, v_2 \in I} \left| B_{v_1} \cap B_{v_2} \right| - \cdots \\
&= \left|B_w\right| - \left| \bigcup_{v \in I} B_v \right| \\
&= \left| B_w \setminus \bigcup_{v \in I} B_v \right|.
\end{align}
This quantity is necessarily non-negative, as desired.
\end{proof}

By setting all $x_i$'s to $1$ in Theorem~\ref{thm:main} we obtain:

\begin{theorem3*} 
Let $w \in S_n$ be a $1432$ and $1423$ avoiding permutation. If $u$ is a subword of $w$, then
\[
\sum_{u \le v \le w} (-1)^{\left|w\right|-\left|v\right|} \Schub_{\perm(v)}(\mathbf{1})\geq 0.
\]
\end{theorem3*}

We conjecture (Conjecture~\ref{conj:1.3}) that Theorem~\ref{cor:main} generalizes to all permutations $w \in S_n$. 

\subsection{Gao's conjecture~\ref{conj:gao}, Theorem~\ref{thm:cw} and its proof}
\label{sec:gao}
Gao~\cite{Gao}
defined a sequence of integers $\{c_u\}_{m \ge 1, u \in S_m}$ recursively, as follows:
\begin{equation} \label{eqn:c_w_defn}
c_w \coloneqq \Schub_w(\mathbf{1}) - 1 - \sum_{\left|u\right| < \left|w\right|} c_u p_u(w),
\end{equation}
where $\left|u\right| = m$ if $u \in S_m$, and $p_u(w)$ is the number of occurrences of $u$ as a pattern in $w$.

Gao
showed that $c_w = 0$ whenever $w(n) = n$, so the definition of $c_w$ can be extended to all $w \in S_\infty$. In the same paper, he conjectured the following:
\begin{conjecture}[{\cite[Conjecture~3.2]{Gao}}] \label{conj:gao2}
We have $c_w \ge 0$ for all $w \in S_\infty$.
\end{conjecture}

Notice that $p_u(w) = \# \{\text{words $v$ such that $u = \perm(v)$ and $v \le w$}\}$. Thus, we can rewrite~\eqref{eqn:c_w_defn} as
\begin{equation} \label{eqn:c_w_defn2}
c_w = \Schub_w(\mathbf{1}) - \sum_{v < w} c_v,
\end{equation}
where the $-1$ has been absorbed into the sum as $c_{()}$. Note that this perspective explains the equivalence of Conjectures~\ref{conj:gao} and~\ref{conj:gao2}.

By inclusion-exclusion, \eqref{eqn:c_w_defn2} is equivalent to
\begin{equation} \label{eqn:c_w_defn3}
c_w = \sum_{v \le w} (-1)^{\left|w\right|-\left|v\right|} \Schub_v(\mathbf{1}).
\end{equation}

Thus, Theorem~\ref{cor:main} immediately implies:

\begin{theorem} \label{thm:conj} 
Conjecture~\ref{conj:gao2} (equivalently, Conjecture~\ref{conj:gao}) holds for $1432$ and $1423$ avoiding permutations $w \in S_{\infty}$. 
\end{theorem} 

Moreover, Theorem~\ref{thm:cw} below provides a combinatorial interpretation for $c_w$ when $w$ is $1432$ and $1423$ avoiding. 

\begin{defi}
Given diagrams $C,D\subseteq [n]\times[n]$ and $k,l\in [n]$, let $\hc_{k,l}$ and $\hd_{k,l}$ denote the diagrams obtained from $C$ and $D$ by removing any boxes in row $k$ or column~$l$. When the indexes $k, l$ are clear from the context we simply write $\hc$ and $\hd$ in place of $\hc_{k,l}$ and $\hd_{k,l}$. 
Fix a diagram $D$. For each diagram $\hc$, let its \emph{augmentation} with respect to the diagram $D$ be:
\[
{\hc}_{\aug}=\hc \cup \{(k,i) \mid (k,i)\in D\} \cup \{(i,l) \mid (i,l) \in D \}\subseteq [n]\times [n].
\] 
\end{defi}

By tracing the proof of Theorem~\ref{thm:main} for the case $u = ()$, we can obtain an interpretation of the coefficient of $\mathbf{m}$ in
$\sum_{v \le w} (-1)^{\left|w\right| - \left|v\right|} M_{w, v}\Schub_v(\mathbf{x}_{w^{-1}(v)})$ in terms of augmentations of diagrams $\widehat{C} \le \widehat{D(w)}$. In particular, we readily obtain:

\begin{theorem2*} 
For $1432$ and $1423$ avoiding permutations $w \in S_{\infty}$ the value of $c_w$ is the number of diagrams $C \le D(w)$ that cannot be written as $\widehat{C}_{\aug}$ for some $\widehat{C}=\widehat{C}_{k, w_k} \le \widehat{D(w)}=\widehat{D(w)}_{k, w_k}$, $k \in \mathbb{Z}_{>0}$. 
\end{theorem2*}

We conclude this section by illustrating Theorem~\ref{thm:cw} for permutations $1342$ and $12453$. 

\begin{exam} 
Computation yields $c_{1342}=0$. We have that $D(1342)=\{(2,2), (3,2)\}$ and thus the diagrams 
% $C^1=\{(2,2), (3,2)\}$, $C^2=\{(1,2), (3,2)\}$, $C^3=\{(1,2), (2,2)\}$ are all the diagrams $C\leq D(1342)$. 
$C\leq D(1342)$ are:
\begin{center}
\includegraphics[scale=1]{Figures/exp_3-15.pdf}\;.
\end{center} 
Note that $C^1=\widehat{C^1}_{\aug}$ with respect to $D(1342)$ with $k=3, l=4$ (or with $k=2, l=3$); $C^2=\widehat{C^2}_{\aug}$ with respect to $D(1342)$ with $k=3, l=4$; $C^3=\widehat{C^3}_{\aug}$ with respect to $D(1342)$ with $k=2, l=3$. Thus, the number of diagrams $C \le D(1342)$ that cannot be written as $\widehat{C}_{\aug}$ for some $\widehat{C} \le \widehat{D(1342)}$ is $0$ yielding $c_{1342}=0$.
\end{exam}

\begin{exam} 
Computation yields $c_{12453}=1$. We have that $D(12453)=\{(3,3), (4,3)\}$ and thus the diagrams 
%$C^1 = \{(3,3), (4,3)\}$, $C^2 = \{(2,3), (4,3)\}$, $C^3 = \{(1,3), (4,3)\}$, $C^4 = \{(2,3), (3,3)\}$, $C^5 = \{(1,3), (3,3)\}$, $C^6 = \{(1,3), (2,3)\}$ are all the diagrams $C\leq D(12453)$. 
$C\leq D(12453)$ are:
\begin{center}
\includegraphics[scale=1]{Figures/exp_3-16.pdf}\;.
\end{center}
Note that $C^1=\widehat{C^1}_{\aug}$ with respect to $D(12453)$ with $k=4, l=5$ (or with $k=3, l=4$); $C^2=\widehat{C^2}_{\aug}$ with respect to $D(12453)$ with $k=4, l=5$; $C^3=\widehat{C^3}_{\aug}$ with respect to $D(12453)$ with $k=4, l=5$; $C^4=\widehat{C^4}_{\aug}$ with respect to $D(12453)$ with $k=3, l=4$; $C^5=\widehat{C^5}_{\aug}$ with respect to $D(12453)$ with $k=3, l=4$. Note also that $C^6$ cannot be written as $\widehat{C^6}_{\aug}$ for some $\widehat{C^6} \le \widehat{D(12453)}$. Thus, the number of diagrams $C \le D(12453)$ that cannot be written as $\widehat{C}_{\aug}$ for some $\widehat{C} \le \widehat{D(1342)}$ is $1$ yielding $c_{12453}=1$.
\end{exam}

\section{Proof of Theorem~\ref{thm:gen}}
\label{sec:gen}
The main result of this section is a generalization of Theorems~\ref{thm:fms} and~\ref{thm:gen}: 

\begin{theorem}\label{thm:gen1} 
Fix a diagram $D \subseteq [n] \times [n]$ and let $\widehat{D}$ be the diagram obtained from $D$ by removing any boxes in row $k$ or column $l$. 
If $K \in \Purple_{k, l}(D)$, then
\[
\chi_D(x_1, \dots, x_n) - M(x_1, \dots, x_n) \chi_{\widehat{D}}(x_1, \dots, x_{k-1}, 0, x_{k+1}, \dots, x_n) \in \Z_{\ge0}[x_1, \dots, x_n],
\]
where 
\[
M(x_1, \dots, x_n) = \prod_{(i, j) \in K} x_i.
\]
\end{theorem}

Theorem~\ref{thm:gen} is a special case of Theorem~\ref{thm:gen1} when $D$ is a Rothe diagram of a permutation and $l=\sigma_k$.

We now proceed to define the set of diagrams $\Purple_{k, l}(D)$ used in the statement of Theorem~\ref{thm:gen1} above.

\begin{defi}
Fix a diagram $D \subseteq [n]\times [n]$ and integers $k, l \in [n]$. Define $\purple_{k, l}(D)$ to be the set of boxes $(i, j)$ such that: 
\begin{itemize}
\item there is some $C \le D$ such that $(i, j) \in C$, but
\item there is no $C \le D$ such that $\widehat{C}_{k, l} \le \widehat{D}_{k, l}$ and $(i, j) \in \widehat{C}_{k, l}$.
\end{itemize}
\end{defi}

\begin{defi}
Fix a diagram $D \subseteq [n]\times [n]$ and integers $k, l \in [n]$. Define $\Purple_{k, l}(D)$ to be the smallest set satisfying the following:
\begin{itemize}
\item $D \setminus \widehat{D}_{k, l} \in \Purple_{k, l}(D)$, and
\item if $K \in \Purple_{k, l}(D)$, $K' \le K$ and $K' \subseteq \purple_{k, l}(D)$, then $K' \in \Purple_{k, l}(D)$.
\end{itemize}
\end{defi}

\begin{exam} \label{p1}
Let $D = D(15243)$, $k = 5$ and $l = 3$. Then 
\[
\purple\nolimits_{k, l}(D) = \{(1, 3), (2, 3), (3, 3), (4, 3)\}
\] 
and 
\[
\Purple\nolimits_{k, l}(D) \Mk =\Mk  \{\{(2, 3), (4, 3)\}, \{(1, 3), (4, 3)\}, \{(2, 3), (3, 3)\}, \{(1, 3), (3, 3)\}, \{(1, 3), (2, 3)\}\}.
\] 
As a result, the set of monomials $M$ produced by Theorem~\ref{thm:gen} is 
\[
\{x_2x_4, x_1x_4, x_2x_3, x_1x_3, x_1x_2\}.
\]
In this case, these are \emph{all} the monomials $M$ for which
\begin{multline*}
\chi_D(x_1,x_2,x_3,x_4,x_5) - M(x_1,x_2,x_3,x_4,x_5) \chi_{\widehat{D}}(x_1,x_2,x_3,x_4,0)\\ 
\in \Z_{\ge0}[x_1,x_2,x_3,x_4,x_5].
\end{multline*} 
See Figure~\ref{fig:p1} for an illustration.
\end{exam}

\begin{figure}[htb]\centering
\renewcommand*{\mk}{\mkern -0.85mu}%
\includegraphics[width=0.85\textwidth]{Figures/fig_15243_5_3.pdf}
\caption{%
The diagram in the first row shows the Rothe diagram of the permutation $15243$. The yellow highlighted row and column correspond to removing row indexed $k=5$ and column indexed $l=3$. The boxes with purple boundary are $\purple_{k, l}(D) = \{(1, 3), (2, 3), (3, 3), (4, 3)\}$. The second row of the figure shows $
\Purple\nolimits_{k, l}(D) = \{\mk \{\mk (2,\mk 3\mk ),\mk (\mk 4, \mk 3\mk )\mk \},\mk 
\{\mk (\mk 1,\mk 3\mk ),\mk (\mk 4,\mk 3\mk )\mk \},\mk \{\mk (\mk 2,\mk 3\mk ),\mk (\mk 3,\mk 3\mk )\mk \},\mk \{\mk (\mk 1,\mk 3\mk ),\mk (\mk 3,\mk 3\mk )\mk \},\mk \{\mk (\mk 1,\mk 3\mk ),\mk (\mk 2,\mk 3\mk )\mk \}\mk \}$ along with the corresponding monomials below each diagram.}
\label{fig:p1}
\end{figure}

\begin{exam} \label{p2}
Let $D = D(15243)$ and $k = l = 4$. Then 
\[
\belowdisplayskip0pt
\purple\nolimits_{k, l}(D) = \{(1, 4), (2, 4), (3, 3), (4, 3)\}
\]
and 
\[
\Purple\nolimits_{k, l}(D) = \{\{(2, 4), (4, 3)\}, \{(1, 4), (4, 3)\}, \{(2, 4), (3, 3)\}, \{(1, 4), (3, 3)\}\}.
\] 
As a result, the set of monomials $M$ produced by Theorem~\ref{thm:gen} is 
\[
\{x_2x_4, x_1x_4, x_2x_3, x_1x_3\}.
\] 
However,
\[
\chi_D(x_1,x_2,x_3,x_4,x_5) - (x_1x_2) \chi_{\widehat{D}}(x_1,x_2,x_3,0,x_5) \in \Z_{\ge0}[x_1,x_2,x_3,x_4,x_5],
\]
so in this case the monomials prescribed by Theorem~\ref{thm:gen} are not the only monomials that could work. See Figure~\ref{fig:p2} for an illustration.
\end{exam}

\begin{figure}[htb]\centering
\includegraphics[width=0.67\textwidth]{Figures/fig_15243_4_4.pdf}
\caption{The diagram in the first row shows the Rothe diagram of the permutation $15243$. The yellow highlighted row and column correspond to removing row indexed $k=4$ and column indexed $l=4$. The boxes with purple boundary are $\purple_{k, l}(D) = \{(1, 4), (2, 4), (3, 3), (4, 3)\}$. The second row of the figure shows $\Purple_{k, l}(D) = \{\{(2, 4), (4, 3)\}, \{(1, 4), (4, 3)\}, \{(2, 4), (3, 3)\}, \{(1, 4), (3, 3)\}\}$ along with the corresponding monomials below each diagram.}
\label{fig:p2}
\end{figure}

The following lemma follows immediately from the definitions:
\begin{lemma} \label{lem:aug}
Let $C, D \subseteq [n] \times [n]$ be diagrams, $k, l \in [n]$, and $K \in \Purple_{k, l}(D)$. If $\widehat{C}_{k, l} \le \widehat{D}_{k, l}$, then $\widehat{C}_{k, l} \cup K \le D$ and $\widehat{C}_{k, l} \cap K = \varnothing$.
\end{lemma}

The following lemma generalizes~\cite[Lemma~5.7]{zeroone}:
\begin{lemma} \label{lem:lift}
Fix a diagram $D$, integers $k, l \in [n]$, $K \in \Purple_{k, l}(D)$, and let $\widehat{D}$ denote $\widehat{D}_{k, l}$. Let $\{\widehat{C}^{(i)}\}_{i \in [m]}$ be a set of diagrams with $\widehat{C}^{(i)} \le \widehat{D}$ for each $i$, and denote $\widehat{C}^{(i)} \cup K$ by $C^{(i)}$ for $i \in [m]$. If the polynomials $\left\{\prod_{j \in [n]} \det(Y^{C^{(i)}_j}_{D_j}) \right\}_{i \in [m]}$ are linearly dependent, then so are the polynomials $\left\{\prod_{j \in [n] \setminus \{l\}} \det(Y^{\widehat{C}^{(i)}_j}_{\widehat{D}_j}) \right\}_{i \in [m]}$.
\end{lemma}
\begin{proof}
We are given that
\begin{equation} \label{lem:lift-eqn:1}
\sum_{i \in [m]} c_i \prod_{j \in [n]} \det\left(Y^{C^{(i)}_j}_{D_j}\right) = 0
\end{equation}
for some constants $(c_i)_{i \in [m]} \in \C^m$ not all zero. Since $C^{(i)} = \widehat{C}^{(i)} \cup K$ for $\widehat{C}^{(i)} \le \widehat{D}$ we have that $C^{(i)}_l = K_l$ for every $i \in [m]$. Thus, \eqref{lem:lift-eqn:1}~can be rewritten as
\begin{equation}
\det(Y^{K_l}_{D_l}) \left( \sum_{i \in [m]} c_i \prod_{j \in [n] \setminus \{l\}} \det\left(Y^{C^{(i)}_j}_{D_j}\right) \right) = 0.
\end{equation}

However, since $\det(Y^{K_l}_{D_l}) \ne 0$, we conclude that
\begin{equation} \label{lem:lift-eqn:3}
\sum_{i \in [m]} c_i \prod_{j \in [n] \setminus \{l\}} \det\left(Y^{C^{(i)}_j}_{D_j}\right) = 0.
\end{equation}

First consider the case that the only boxes of $D$ in row $k$ or column $l$ are those in $D_l$. If this is the case then
\begin{equation} 
\prod_{j \in [n] \setminus \{l\}} \det\left(Y^{\widehat{C}^{(i)}_j}_{\widehat{D}_j}\right) = \prod_{j \in [n] \setminus \{l\}} \det\left(Y^{C^{(i)}_j}_{D_j}\right)
\end{equation}
for each $i \in [m]$. Therefore,
\begin{equation} \label{lem:lift-eqn:5}
\sum_{i \in [m]} c_i \prod_{j \in [n] \setminus \{l\}} \det\left(Y^{\widehat{C}^{(i)}_j}_{\widehat{D}_j}\right) = \sum_{i \in [m]} c_i \prod_{j \in [n] \setminus \{l\}} \det\left(Y^{C^{(i)}_j}_{D_j}\right).
\end{equation}

Combining~\eqref{lem:lift-eqn:3} and~\eqref{lem:lift-eqn:5} we obtain that the polynomials 
$\left\{\Mk\Mk \prod_{j \mk\in\mk [n] \mk \setminus \mk \{\mk l\mk \}} 
\Mk \det(Y^{\widehat{C}^{(i)}_j}_{\widehat{D}_j}\Mk ) \Mk \Mk \right\}_{\Mk \Mk i \mk \in\mk [m]}$ 
are linearly dependent, as desired.

Now, suppose that there are boxes of $D$ in row $k$ that are not in $D_l$. Let $j_1 < \dots < j_p$ be all indices $j \ne l$ such that $D_j = \widehat{D}_j \cup \{k\}$. Then, for each $i \in [m]$ and $q \in [p]$, $C^{(i)}_{j_q} \setminus \widehat{C}^{(i)}_{j_q} = K_{j_q}$. For each $q \in [p]$, let $k_q$ be the only element of $K_{j_q}$; then~\eqref{lem:lift-eqn:3} implies that
\begin{equation} \label{lem:lift-eqn:6}
\left[ \prod_{q \in [p]} y_{k_q k} \right] \sum_{i \in [m]} c_i \prod_{j \in [n] \setminus \{l\}} \det\left(Y^{C^{(i)}_j}_{D_j}\right) = 0.
\end{equation}

However,
\begin{equation}
\left[ \prod_{q \in [p]} y_{k_q k} \right] \prod_{j \in [n] \setminus \{l\}} \det\left(Y^{C^{(i)}_j}_{D_j}\right) = \prod_{j \in [n] \setminus \{l\}} \det\left(Y^{\widehat{C}^{(i)}_j}_{\widehat{D}_j}\right),
\end{equation}
as is seen by Laplace expansion on the $k_q$th row of $\det(Y^{C^{(i)}_{j_q}}_{D_{j_q}})$, and therefore
\begin{equation} \label{lem:lift-eqn:8}
\left[ \prod_{q \in [p]} y_{k_q k} \right] \sum_{i \in [m]} c_i \prod_{j \in [n] \setminus \{l\}} \det\left(Y^{C^{(i)}_j}_{D_j}\right) = \sum_{i \in [m]} c_i \prod_{j \in [n] \setminus \{l\}} \det\left(Y^{\widehat{C}^{(i)}_j}_{\widehat{D}_j}\right).
\end{equation}

Thus, \eqref{lem:lift-eqn:6} and~\eqref{lem:lift-eqn:8} imply that
\begin{equation}
\sum_{i \in [m]} c_i \prod_{j \in [n] \setminus \{l\}} \det\left(Y^{\widehat{C}^{(i)}_j}_{\widehat{D}_j}\right) = 0,
\end{equation}
as desired.
\end{proof}

We are now ready to prove Theorem~\ref{thm:gen1}:
%\bigskip

%\noindent
\begin{proof}[Proof of Theorem~\ref{thm:gen1}] 
Let $M = M(x_1, \dots, x_n)$.
Suppose there is some $K \in \Purple_{k, l}(D)$ such that
\[
M(x_1, \dots, x_n) = \prod_{(i, j) \in K} x_i.
\]

We must show that $[M\mathbf{m}] \chi_D \ge [\mathbf{m}] \chi_{\widehat{D}}$ for each monomial $\mathbf{m}$ of $\chi_{\widehat{D}}$ not divisible by $x_k$. Let $\widehat{\mathcal{C}}$ be the set of diagrams $\widehat{C}$ such that $\widehat{C} \le \widehat{D}$ and $\prod_{(i, j) \in \widehat{C}} x_i = \mathbf{m}$. By Corollary 5.5,
\[
[\mathbf{m}]\chi_{\widehat{D}} = \dim \left( \textstyle \Span_\C \displaystyle \left\{ \prod_{j=1}^n \det\left(Y^{\widehat{C}_j}_{\widehat{D}_j}\right) \;\middle|\; \widehat{C} \in \widehat{\mathcal{C}} \right\} \right).
\]

Let $\mathcal{C} = \{ \widehat{C} \cup K \mid \widehat{C} \in \widehat{\mathcal{C}} \}$. By Lemma~\ref{lem:aug}, every $C \in \mathcal{C}$ satisfies $C \le D$ and $\prod_{(i, j) \in C} x_i = M\mathbf{m}$, so Corollary 5.5 implies that
\[
[M\mathbf{m}]\chi_D \ge \dim \left( \textstyle \Span_\C \displaystyle \left\{ \prod_{j=1}^n \det\left(Y^{C^{(i)}_j}_{D_j}\right) \;\middle|\; C \in \mathcal{C} \right\} \right).
\]
Note the inequality, which is because we have only a subset of the $C$.

Finally, Lemma~\ref{lem:lift} implies that
\[
\dim \left( \Mk \Mk \textstyle \Span_\C \Mk \displaystyle \left\{ \prod_{j=1}^n \det\Mk \left(\Mk Y^{C^{(i)}_j}_{D_j}\Mk \right) \;\middle|\; C \in \mathcal{C} \right\} \Mk \right) \Mk\ge\Mk \dim\Mk \Mk  \left(\Mk \Span\nolimits_\C \displaystyle \left\{ \prod_{j=1}^n \det\Mk \left(\Mk Y^{\widehat{C}^{(i)}_j}_{\widehat{D}_j}\Mk \right) \,\middle|\, \widehat{C} \in \widehat{\mathcal{C}} \right\} \Mk \right)\Mk,
\]
so $[M\mathbf{m}]\chi_D \ge [\mathbf{m}]\chi_{\widehat{D}}$ for each monomial $\mathbf{m}$ of $\chi_{\widehat{D}}$ not divisible by $x_k$.
\end{proof}

\section{Questions and Conjectures}
\label{sec:conj}

The results of this paper naturally give rise to the following Conjectures and Questions. 

\subsection{Extending Theorem~\ref{cor:main}} 
\begin{conjecture} \label{conj:1.3}
Let $w \in S_n$. If $u$ is a subword of $w$, then
\[
\sum_{u \le v \le w} (-1)^{\left|w\right|-\left|v\right|} \Schub_{\perm (v)}(\mathbf{1})\geq 0.
\]
\end{conjecture}

Theorem~\ref{cor:main} confirms the above conjecture for $1432$ and $1423$ avoiding permutations. A special case of Conjecture~\ref{conj:1.3} is Gao's conjecture~\ref{conj:gao}. Conjecture~\ref{conj:1.3} has been verified by computer for all permutations in $S_n$ for $n\leq 8$.

\subsection{Extending Theorem~\ref{thm:gen}} 
The monomials $M(x_1, \dots, x_n) = \prod_{(i, j) \in K} x_i$ we constructed from 
diagrams $K \in \Purple_{k, \sigma_k}(D(\sigma))$ in Theorem~\ref{thm:gen} do not always characterize all monomials for which 
\[
\mathfrak{S}_{\sigma}(x_1, \ldots, x_n)-M(x_1, \ldots, x_n) \mathfrak{S}_{\pi}(x_{1}, \ldots, \widehat{x_k}, \ldots, x_{n}) \in \mathbb{Z}_{\geq 0}[x_1, \ldots, x_n]
\] 
holds; recall that $\pi \in S_{n-1}$ is obtained by removing row $k$ and column $\sigma_k$ of $D(\sigma)$. The following example illustrates this:


\begin{exam}
For the permutation $\sigma=1432$ and its pattern $\pi=132$ (coming from the subword $142$ of $1432$) obtained by removing row $k=3$ and column $\sigma_k=3$ of $D(\sigma)$, the set of monomials of the form $\prod_{(i, j) \in K} x_i$ constructed from diagrams $K \in \Purple_{3, \sigma_3}(D(\sigma))$ is $\{x_1x_3, x_2x_3\}$, yet the monomial $M(x_1, \ldots, x_n) = x_1x_2$ also yields $\mathfrak{S}_{\sigma}(x_1, \ldots, x_n)-M(x_1, \ldots, x_n) \mathfrak{S}_{\pi}(x_{1}, \ldots, \widehat{x_k}, \ldots, x_{n}) \in \mathbb{Z}_{\geq 0}[x_1, \ldots, x_n]$. In contrast, for $\sigma=1432$ and its pattern $\pi=132$ (coming from the subword $143$ of $1432$) obtained by removing row $k=4$ and column $\sigma_4=2$ of $D(\sigma)$, the set of monomials of the form $\prod_{(i, j) \in K} x_i$ constructed from diagrams $K \in \Purple_{4, \sigma_4}(D(\sigma))$ is $\{ x_1x_2, x_1x_3, x_2x_3\}$ and these are all the monomials for which $\mathfrak{S}_{\sigma}(x_1, \ldots, x_n)-M(x_1, \ldots, x_n) \mathfrak{S}_{\pi}(x_{1}, \ldots, \widehat{x_k}, \ldots, x_{n}) \in \mathbb{Z}_{\geq 0}[x_1, \ldots, x_n]$. 
\end{exam}



\begin{ques} \label{prob:char} 
Given permutation $\sigma \in S_n$ and its pattern $\pi \in S_{n-1}$ obtained by removing row $k$ and column $\sigma_k$ of $D(\sigma)$, characterize all monomials $M(x_1, \ldots, x_n)$ for which 
\[
\mathfrak{S}_{\sigma}(x_1, \ldots, x_n)-M(x_1, \ldots, x_n) \mathfrak{S}_{\pi}(x_{1}, \ldots, \widehat{x_k}, \ldots, x_{n}) \in \mathbb{Z}_{\geq 0}[x_1, \ldots, x_n]
\]
holds. 
\end{ques}

We conjecture that for $1432$ and $1423$ avoiding permutations Theorem~\ref{thm:gen} characterizes these monomials:

\begin{conjecture} \label{conj:16} 
For $1432$ and $1423$ avoiding permutation $\sigma \in S_n$ and its pattern $\pi \in S_{n-1}$ obtained by removing row $k$ and column $\sigma_k$ of $D(\sigma)$, 
\[
\mathfrak{S}_{\sigma}(x_1, \ldots, x_n)-M(x_1, \ldots, x_n) \mathfrak{S}_{\pi}(x_{1}, \ldots, \widehat{x_k}, \ldots, x_{n}) \in \mathbb{Z}_{\geq 0}[x_1, \ldots, x_n]
\] 
if and only if $M(x_1, \dots, x_n) = \prod_{(i, j) \in K} x_i$, where $K \in \Purple_{k, \sigma_k}(D(\sigma))$. 
\end{conjecture}

Conjecture~\ref{conj:16} has been verified by computer for all permutations in $S_n$ for $n\leq 8$.
We note that there are permutation and pattern pairs $\sigma \in S_n$ and $\pi \in S_{n-1}$, where $\sigma$ is not $1432$ and $1423$ avoiding, yet $\mathfrak{S}_{\sigma}(x_1, \ldots, x_n)-M(x_1, \ldots, x_n) \mathfrak{S}_{\pi}(x_{1}, \ldots, \widehat{x_k}, \ldots, x_{n}) \in \mathbb{Z}_{\geq 0}[x_1, \ldots, x_n]$ if and only if $M(x_1, \dots, x_n) = \prod_{(i, j) \in K} x_i$, where $K \in \Purple_{k, \sigma_k}(D(\sigma))$. An example is $\sigma=1423$ and any of its patterns $\pi$ obtained from $D(\sigma)$ by removing row $k$ and column $\sigma_k$ ($k \in [4]$).


\subsection{Extending Theorem~\ref{thm:main}} 
As stated, Theorem~\ref{thm:main} does not hold for all permutations. However, it is natural to wonder about the following extension:

\begin{ques} \label{prob:gen} 
Let $w \in S_n$ and let $u$ be a subword of $w$. Using the monomials from Theorem~\ref{thm:gen} \textup{(}or its extension asked for in Problem~\ref{prob:char}\textup{)} is it possible to pick suitable monomials $m_{w,v} \in \Z[x_1, \ldots, x_n]$ to make the expression
\[
\sum_{u \le v \le w} (-1)^{\left|w\right|-\left|v\right|} m_{w, v} \Schub_{\perm(v)}(\mathbf{x}_{w^{-1}(v)})
\] 
belong to $\Z_{\ge0}[x_1, \dots, x_n]$?
\end{ques}

Note that a positive answer to Question~\ref{prob:gen} would be an extension of Theorem~\ref{thm:main} which would readily imply Conjecure~\ref{conj:1.3} as well as Gao's Conjecture~\ref{conj:gao}.

\bibliographystyle{amsplain-ac}
\bibliography{ALCO_Meszaros_592}
\end{document}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
