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\title[Double orthodontia formulas and Lascoux positivity]{Double orthodontia formulas\texorpdfstring{\\}{} and Lascoux positivity}

\author{\firstname{Linus} \lastname{Setiabrata}}
\address{Massachusetts Institute of Technology\\ 
Department of Mathematics\\
Cambridge, MA 02139}
\email{setia@mit.edu}

\author{\firstname{Avery}  \lastname{St.\ Dizier}}
\address{Michigan State University\\
Department of Mathematics\\
East Lansing, MI 48824}
\email{stdizier@msu.edu}

\keywords{Grothendieck polynomials, Lascoux polynomials}
  
\subjclass{05E05, 05E10}

\begin{document}

\begin{abstract}
We give a new formula for double Grothendieck polynomials based on Magyar's orthodontia algorithm for diagrams. Our formula implies a similar formula for double Schubert polynomials $\mathfrak S_w(\mathbf x;\mathbf y)$. We also prove a curious positivity result: for vexillary permutations $w\in S_n$, the polynomial $x_1^n\dots x_n^n \mathfrak S_w(x_n^{-1}, \dots, x_1^{-1}; 1,\dots,1)$ is a graded nonnegative sum of Lascoux polynomials. We conjecture that this positivity result holds for all $w\in S_n$. This conjecture would follow from a problem of independent interest regarding Lascoux positivity of certain products of Lascoux polynomials.
\end{abstract}

\maketitle

\section{Introduction}
\subsection{Orthodontia and flagged Weyl modules}
The Schubert polynomials, denoted $\mathfrak S_w(x_1, \dots, x_n)$ and introduced by Lascoux and Sch\"utzenberger~\cite{ls82}, are distinguished representatives of Schubert varieties in the cohomology of the flag variety of~$\CC^n$. These polynomials have very rich combinatorial structure~\cite{bb93,bjs93,fk96,km05,lls21,hmms22} and play a central role in algebraic combinatorics.

Schubert polynomials $\mathfrak S_w(\mathbf x)$ are indexed by permutations $w \in S_n$. The Rothe diagram $D(w)$ of $w\in S_n$ encodes a plethora of information about $\mathfrak S_w(\mathbf x)$. More generally, for any \%-avoiding diagram $D$, there is a representation of the group $B$ of invertible upper triangular matrices called the \emph{flagged Weyl module} $\mathcal M_D$. When $D = D(w)$ is a Rothe diagram, the dual character $\chi_D$ of the representation $\mathcal M_D$ is equal to the Schubert polynomial $\mathfrak S_w(\mathbf x)$~\cite{kp87,kp04}. The study of dual characters $\chi_D$ as a whole has shed light on Schubert and related polynomials~\cite{fms18,hmss24}.

Using a geometric interpretation of $\mathcal M_D$ as the space of sections of a certain line bundle on a variety, Magyar~\cite{magyar98} showed that $\chi_D$ is given by the formula
\[
\chi_D = \omega_1^{k_1}\dots\omega_n^{k_n}\pi_{i_1}(\omega_{i_1}^{m_1}\pi_{i_2}(\omega_{i_2}^{m_2}(\dots\pi_{i_\ell}(\omega_{i_\ell}^{m_\ell})\dots))).
\]
Here, $\omega_i = x_1\dots x_i$ is a fundamental weight, $\pi_i = \del_i x_i$ is a Demazure operator, and
\[
\mathbf k(D) = (k_1,\dots, k_n), \qquad \mathbf i(D) = (i_1, \dots, i_\ell), \qquad \mathbf m(D) = (m_1, \dots, m_\ell)
\]
is combinatorial data associated to the \emph{orthodontic sequence} of $D$, which builds a \%-avoiding diagram from ``smaller'' \%-avoiding diagrams.

\subsection{Doubled orthodontia}
The double Grothendieck polynomials, denoted $\mathfrak G_w(x_1, \dots, x_n; y_1, \dots, y_n)$ and introduced by Lascoux~\cite{lascoux90}, are distinguished representatives of Schubert varieties in the equivariant $K$-theory of the flag variety. Schubert polynomials can be obtained from double Grothendieck polynomials by setting $y_i\mapsto 0$ (corresponding to forgetting equivariance) and taking the lowest degree part (corresponding to taking the associated graded in $K$-theory). There is considerable interest in extending our understanding of Schubert polynomials to double Grothendieck polynomials and their various specializations~\cite{fk94, km04,sam11, weigandt21, lls23, ccmm23,bfhtw23, psw24, hmss24}.

There is no known $K$-theoretic analogue of $\mathcal M_D$. Despite this, we can combinatorially extend Magyar's formula to double Grothendieck polynomials:
\begin{thm}
\label{thm:double-groth-master}
Let $D$ be a \%-avoiding diagram with double orthodontic sequence $\mathbf K, \mathbf i, \mathbf j, \mathbf M$. Define
\begin{equation}
\label{eqn:double-groth-master}
\mathscr G_D(\mathbf x;\mathbf y)\colonequals \overline\omega_1^{K_1}\overline\omega_2^{K_2}\dots\overline\omega_n^{K_n}\overline\pi_{i_1,j_1}(\overline\omega_{i_1}^{M_1}\overline\pi_{i_2,j_2}(\overline\omega_{i_2}^{M_2}\dots\overline\pi_{i_\ell,j_\ell}(\overline\omega_{i_\ell}^{M_\ell})\dots)).
\end{equation}
When $D = D(w)$ is the Rothe diagram of a permutation, then $\mathscr G_D(\mathbf x; \mathbf y) = \mathfrak G_w(\mathbf x; \mathbf y)$.
\end{thm}
The operators $\overline\omega_i^M$ and $\overline\pi_{i,j}$ are defined in Equations~\eqref{eqn:pi-ij-operators} and~\eqref{eqn:omega-iM-operators}.

Double Schubert polynomials $\mathfrak S_w(\mathbf x; \mathbf y)$ are equal to the lowest degree part of $\mathfrak G_w(\mathbf x; -\mathbf y)$, and Theorem~\ref{thm:double-groth-master} implies a similar formula for double Schubert polynomials (Corollary~\ref{cor:double-schub-master}). Theorem~\ref{thm:double-groth-master} also extends our previous joint work with M\'esz\'aros~\cite[Thm 1.1]{mss22}, which gave a similar formula for ordinary Grothendieck polynomials.

As is the case with Magyar's formula for $\chi_D$, and in contrast to the usual degree-\emph{decreasing} recurrence defining Grothendieck polynomials via divided difference operators, the formula~\eqref{eqn:double-groth-master} is degree-\emph{increasing}. This makes the formula~\eqref{eqn:double-groth-master} well suited for some combinatorial applications.

\subsection{Lascoux positivity}
When $D = D(\alpha)$ is the skyline diagram of a composition $\alpha\in\NN^n$, the dual character $\chi_D$ is the \emph{key polynomial} $\kappa_\alpha(x_1, \dots, x_n)$~\cite{rs95}. Key polynomials are minimal among the dual characters $\chi_D$ of \%-avoiding diagrams: every dual character, and in particular every Schubert polynomial, is a nonnegative sum of key polynomials~\cite{rs98}.

The Lascoux polynomials $\mathfrak L_\alpha(x_1, \dots, x_n)$, indexed by compositions $\alpha\in\NN^n$, are inhomogeneous polynomials which often play the same role to the key polynomials as Grothendieck polynomials do to Schubert polynomials. One inhomogeneous extension of the key positivity~\cite{rs98} of Schubert polynomials is the Grothendieck-to-Lascoux expansion~\cite{ry21,sy23}. One application of our formula~\eqref{eqn:double-groth-master} is another inhomogeneous extension of this key positivity:

\begin{thm}
\label{thm:lascoux-positivity-main}
Let $D\subseteq[n]\times[m]$ be a diagram whose columns are ordered by inclusion. Let $\mathscr S_D(\mathbf x; \mathbf y)$ be the lowest degree part of $\mathscr G_D(\mathbf x; \mathbf y)$. Then, the polynomial
\[
x_1^m \dots x_n^m \mathscr S_D(x_n^{-1}, \dots, x_1^{-1};  -1, \dots, -1)
\]
is a graded nonnegative sum of Lascoux polynomials $\mathfrak L_\alpha(x_1, \dots, x_n)$.
\end{thm}
Theorem~\ref{thm:lascoux-positivity-main} can be modified to incorporate $\beta$-Lascoux polynomials: the $\beta$-Lascoux expansion of $x_1^m\dots x_n^m \mathscr S_D(x_n^{-1}, \dots, x_1^{-1}; \beta,\dots,\beta)$ is $\ZZ_{\geq 0}[\beta]$-nonnegative.

Theorem~\ref{thm:lascoux-positivity-main} and Corollary~\ref{cor:double-schub-master} together imply the following corollary.
\begin{coro}
\label{cor:lascoux-positivity-schub}
Let $w\in S_n$ be a vexillary permutation and denote the corresponding double Schubert polynomial by $\mathfrak S_w(x_1, \dots, x_n; y_1, \dots, y_n)$. Then the polynomial
\[
x_1^n \dots x_n^n \mathfrak S_w(x_n^{-1}, \dots, x_1^{-1}; 1, \dots, 1)
\]
is a graded nonnegative sum of Lascoux polynomials $\mathfrak L_\alpha(x_1, \dots, x_n)$.
\end{coro}

We conjecture that the extra assumption on the diagram $D$ in Theorem~\ref{thm:lascoux-positivity-main} is unnecessary.
\begin{conj}
\label{conj:lascoux-positivity-general}
For any \%-avoiding diagram $D\subseteq[n]\times[m]$, the polynomial
\[
x_1^m \dots x_n^m \mathscr S_D(x_n^{-1}, \dots, x_1^{-1}; -1, \dots, -1)
\]
is a graded nonnegative sum of Lascoux polynomials $\mathfrak L_\alpha(x_1, \dots, x_n)$. In particular, the same holds for
\[
x_1^n\dots x_n^n\mathfrak S_w(x_n^{-1}, \dots, x_1^{-1}; 1, \dots, 1),
\]
for any $w \in S_n$.
\end{conj}
In Proposition~\ref{prop:mult-implies-main}, we show that Conjecture~\ref{conj:lascoux-positivity-general} would follow from the following conjecture of independent interest:

\begin{conj}
\label{conj:mult-pos}
The product $x_1\dots x_i(1-x_{i+1})\dots(1-x_n)\mathfrak L_\alpha$, for any $\alpha\in\NN^n$ and $i\in[n]$, is a graded nonnegative linear combination of Lascoux polynomials.
\end{conj}

\subsection{Outline of the paper}
Our proof of Theorem~\ref{thm:double-groth-master} is based on an improvement on the ideas in~\cite{mss22}: we induct on a partial order on $S_n$ called the \emph{orthodontic sort order} (Definition~\ref{defn:os}), and a key step is to give a precise description of the first few terms of the orthodontic sequences of \emph{sorted permutations} (Theorem~\ref{thm:os2-orth}). Although the argument is narratively similar to that in~\cite{mss22}, we need different methods.

Our proof of Theorem~\ref{thm:lascoux-positivity-main} examines the orthodontic sequence of diagrams $D$ whose columns are totally ordered by inclusion (Corollary~\ref{cor:omegas-then-pis}) and studying the behavior of the operators $\omega_i^M$ and $\pi_{i,j}$ when specializing $y_j\mapsto -1$ and intertwining with the operator $r_{m,n}$ of~\cite[Defn 1.1] {yu23} defined on polynomials by
\[
f(x_1, \dots, x_n)\mapsto x_1^m \dots x_n^m f(x_n^{-1}, \dots, x_1^{-1}).
\]

\section{Background}
\subsection{Conventions}
We write permutations $w \in S_n$ in one-line notation. For $j\in[n-1]$, the notation $s_j$ denotes the adjacent transposition in $S_n$ which swaps $j$ and~$j+1$. For $w\in S_n$, let $\ell(w)$ denote the length of $w$. Permutations act on the right: $ws_j$ is equal to $w$ with $w(j)$ and $w(j+1)$ swapped.
\subsection{Difference operators and families of polynomials}
For $i \in [n-1]$, define the \emph{divided difference operator} $\del_i\colon \RR[x_1, \dots, x_n]\to \RR[x_1, \dots, x_n]$ by the formula
\[
\del_i(f) \colonequals \frac{f - s_i f}{x_i - x_{i+1}}.
\]
The \emph{isobaric divided difference operators} $\overline\del_i$, \emph{Demazure operators} $\pi_i$, and \emph{Demazure--Lascoux operators} $\overline\pi_i$ are defined on $\RR[x_1, \dots, x_n]$ by the formulas
\begin{align*}
\overline\del_i(f) &\colonequals \del_i((1 - x_{i+1})f),\\
\pi_i(f) &\colonequals \del_i(x_if),\\
\overline\pi_i(f) &\colonequals \overline\del_i(x_if).
\end{align*}
\begin{lemm}
\label{lem:varphi-braid}
Let $\varphi_i$ denote any of $\del_i$, $\overline\del_i$, $\pi_i$, or $\overline\pi_i$. Then
\[
\varphi_i \varphi_{i+1}\varphi_i = \varphi_{i+1}\varphi_i\varphi_{i+1} \qquad\textup{ and } \qquad \varphi_i\varphi_j = \varphi_j\varphi_i\textup{ if } |i-j| \geq 2.
\]
\end{lemm}
The \emph{double Grothendieck polynomial} $\mathfrak G_w(\mathbf x; \mathbf y)$ of $w\in S_n$ is defined recursively on the weak Bruhat order, starting from the longest permutation $w_0 \in S_n$. The polynomial $\mathfrak G_w(\mathbf x; \mathbf y)$ is defined by
\[
\mathfrak G_w(\mathbf x; \mathbf y) = \begin{cases} \prod_{i+j\leq n}(x_i + y_j - x_iy_j) &\textup{ if } w = w_0,\\ \overline\del_i\mathfrak G_{ws_i}(\mathbf x; \mathbf y)&\textup{ if } \ell(w) < \ell(ws_i).\end{cases}
\]
Lemma~\ref{lem:varphi-braid} guarantees that double Grothendieck polynomials are well-defined. The \emph{double Schubert polynomial} $\mathfrak S_w(\mathbf x; \mathbf y)$ is defined to be the lowest degree part of $\mathfrak G_w(\mathbf x; -\mathbf y)$. In particular,
\[
\mathfrak S_w(\mathbf x; \mathbf y) = \begin{cases} \prod_{i+j\leq n}(x_i - y_j) &\textup{ if } w = w_0,\\ \del_i\mathfrak S_{ws_i}(\mathbf x; \mathbf y) &\textup{ if } \ell(w) < \ell(ws_i).\end{cases}
\]
The ordinary \emph{Grothendieck polynomials} $\mathfrak G_w(\mathbf x)$ and ordinary \emph{Schubert polynomials} $\mathfrak S_w(\mathbf x)$ are defined to be the specializations $\mathfrak G_w(\mathbf x; \mathbf 0)$ and $\mathfrak S_w(\mathbf x;\mathbf 0)$ of their doubled counterparts. In particular,
\begin{align*}
\mathfrak G_w(\mathbf x) &= \begin{cases} x_1^n x_2^{n-1} \dots x_n &\textup{ if } w = w_0,\\ \overline\del_i\mathfrak G_{ws_i}(\mathbf x) &\textup{ if } \ell(w) < \ell(ws_i),\end{cases} \qquad \textup{ and } \\
\mathfrak S_w(\mathbf x) &= \begin{cases} x_1^n x_2^{n-1}\dots x_n &\textup{ if } w = w_0,\\ \del_i\mathfrak S_{ws_i}(\mathbf x) &\textup{ if } \ell(w) < \ell(ws_i).\end{cases}
\end{align*}
The symmetric group $S_n$ acts on compositions via permuting coordinates. The \emph{Lascoux polynomial} $\mathfrak L_\alpha$ of a composition $\alpha \in \NN^n$ is defined recursively by
\[
\mathfrak L_\alpha(\mathbf x) = \begin{cases} x_1^{\alpha_1}\dots x_n^{\alpha_n}&\textup{ if } \alpha_1 \geq \dots \geq \alpha_n\\ \overline\pi_i\mathfrak L_{\alpha\cdot s_i}(\mathbf x) &\textup{ if } \alpha_i < \alpha_{i+1}.\end{cases}
\]
Lemma~\ref{lem:varphi-braid} guarantees that Lascoux polynomials are well-defined. The \emph{key polynomial} $\kappa_\alpha(\mathbf x)$ is defined to be the lowest degree part of $\mathfrak L_\alpha(\mathbf x)$. In particular,
\[
\kappa_\alpha(\mathbf x) = \begin{cases} x_1^{\alpha_1}\dots x_n^{\alpha_n}&\textup{ if } \alpha_1 \geq \dots \geq \alpha_n\\ \pi_i\kappa_{\alpha\cdot s_i}(\mathbf x) &\textup{ if } \alpha_i < \alpha_{i+1}.\end{cases}
\]
\subsection{Pipe dreams}
A \emph{pipe dream} is a filling of a triangular grid $\{(i,j)\in[n]\times[n]\colon i + j \leq n\}$ with crossing tiles \includegraphics[scale=0.5]{figures/crossing-tile} and bumping tiles \includegraphics[scale=0.5]{figures/bumping-tile}. By placing half-bumping tiles \includegraphics[scale=0.5]{figures/half-bump-tile} at $\{(i,j)\in[n]\times[n]\colon i+j = n+1\}$, a pipe dream forms a network of $n$ pipes running from the top edge of the grid to the left edge (see Figure~\ref{fig:1423-pd}).

For any pipe dream $P$, there is an associated permutation $\del(P) \in S_n$ given by labeling the pipes $1$ through $n$ along the top edge, tracing the pipes and ignoring any crossings between any pair of pipes which have already crossed, i.e.\ replacing redundant crossing tiles with bump tiles; reading the labels of the pipes along the left edge from top to bottom gives a string of numbers $(\del(P))(1), (\del(P))(2), \dots, (\del(P))(n)$ that defines $\del(P) \in S_n$. (The permutation $\del(P)$ is the \emph{Demazure product} of the transpositions $s_i\in S_n$ corresponding to antidiagonals on which the crosses sit, reading right to left, starting from the top row; cf.~\cite[Ex 5.1]{km04},~\cite[\S 6.1]{weigandt21}. Our convention agrees with the original definition~\cite{bb93}: the pipe at the $i$\textsuperscript{th} row is connected to the $(\del(P))(i)$\textsuperscript{th} column.)

Identify $P$ with the set $\{(i,j) \colon P\textup{ has } \includegraphics[scale=0.5]{figures/crossing-tile} \textup{ at } (i,j)\}$ of crosses of $P$, and denote by $P^{(i)}$ the pipe of $P$ entering in the $i$\textsuperscript{th} column.

\begin{example}
\label{ex:1423-pd}
All five pipe dreams $P$ such that $\del(P) = 1423$ are displayed in Figure~\ref{fig:1423-pd}. Redundant crossing tiles \includegraphics[scale=0.5]{figures/crossing-tile} are highlighted in red.
\begin{figure}[ht]
\centering
\includegraphics[scale=0.7]{figures/1423-pd}
\caption{The five pipe dreams in $\PD(1423)$.}
\label{fig:1423-pd}
\end{figure}
\end{example}

\begin{thm}[{\cite[Thm 6.1]{weigandt21}; see also~\cite[Cor 5.4]{km04},~\cite[Thm 2.3]{fk94}}]
The double Grothendieck polynomial of $w\in S_n$ is given by the formula
\[
\mathfrak G_w(\mathbf x; \mathbf y) = \sum_{P \in \PD(w)}\prod_{(i,j) \in P}(-1)^{|P| - \ell(w)} (x_i + y_j - x_iy_j).
\]
\end{thm}

\subsection{Orthodontic sequence}
A \emph{diagram} is defined to be a subset $D\subseteq [n]\times[m]$. View $D = (D_1, \dots, D_m)$ as a subset of an $n\times m$ grid, where $D_j\colonequals \{i \in[n]\colon (i,j)\in D\}$ encodes the $j$\textsuperscript{th} column: an element $i \in D_j$ corresponds to a box in row~$i$ and column~$j$.

\begin{defi}
Let $w \in S_n$. The Rothe diagram $D(w)$ is defined to be
\[
D(w) = \{(i,j)\in[n]\times[n]\colon i < w^{-1}(j) \textup{ and } j < w(i)\}.\qedhere
\]
\end{defi}

\begin{example}
The Rothe diagram $D(31542)$ consists of the purple squares in Figure~\ref{fig:31542-rothe}.
\begin{figure}[ht]
\centering
\includegraphics{figures/31542-rothe}
\caption{The Rothe diagram for $w = 31542$.}
\label{fig:31542-rothe}
\end{figure}
\end{example}

\begin{defi}[\cite{rs98}]
A diagram $D$ is called \emph{\%-avoiding} when $i_2 \in D_{j_1}$ and $i_1 \in D_{j_2}$ for $i_1 < i_2$ and $j_1 < j_2$ implies $i_1 \in D_{j_1}$ or $i_2 \in D_{j_2}$.
\end{defi}

Equivalently, a diagram is \%-avoiding if it does not have a pair of rows and a pair of columns to which its restriction looks like the configuration in Figure~\ref{fig:percentage_avoiding}.

\begin{figure}[ht]
	\begin{center}
		\includegraphics{figures/percentage-avoiding.pdf}
	\end{center}
	\caption{A forbidden configuration in a \%-avoiding diagram.}
	\label{fig:percentage_avoiding}
\end{figure}

Given a \%-avoiding diagram $D$, we describe an algorithm to compute its \emph{double orthodontic sequence}
\[
\mathbf K(D) \!=\! (K_1, \dots, K_n), \hspace{0.4em} \mathbf i(D) \!=\! (i_1, \dots, i_\ell), \hspace{0.4em} \mathbf j(D) \!=\! (j_1, \dots, j_\ell), \hspace{0.4em} \mathbf M(D) \!=\! (M_1, \dots, M_\ell).
\]

Begin by setting $K_i \colonequals \{j\colon D_j = [i]\}$, and let $D_-$ denote the diagram obtained by replacing any such columns with the empty column. If $D_-$ is empty, then $\mathbf i = \mathbf j = \mathbf M$ is the empty vector and the algorithm terminates.

An \emph{orthodontic step} applied to $D_-$ outputs a diagram $(D_-)''$ which we now describe. A \emph{missing tooth} of a column $C\subseteq[n]$ is an integer $i\in[n]$ so that $i\not\in C$ and $i+1\in C$; any nonempty column of $D_-$ has a missing tooth. Set $i_1$ to be the smallest missing tooth of the leftmost nonempty column $(D_-)_j$ of $D_-$. Set $j_1$ to be $j - \#\{a\leq i_1\colon a\not\in (D_-)_j\}$. Now swap rows $i_1$ and $i_1+1$ to get a diagram $(D_-)'$. Any column of $(D_-)'$ which is a standard interval is necessarily equal to $[i_1]$; set $M_1 = \{j\colon (D_-)'_j = [i_1]\}$. Let $(D_-)''$ denote the diagram obtained from $(D_-)'$ by removing all columns equal to $[i_1]$. 

\begin{lemm}[{\cite[Prop 12]{rs98}, see also~\cite[pg.\ 12]{magyar98}}]
Any \%-avoiding diagram $D$ will be the empty diagram after sufficiently many orthodontic steps.
\end{lemm}

Repeatedly apply orthodontic steps to the resulting diagrams $(D_-)''$, keeping track of the data $\mathbf i, \mathbf j, \mathbf M$ at each step, until the diagram is empty. These diagrams form the double orthodontic sequence of $D$.

\begin{example}
The orthodontic sequence for the Rothe diagram $D(31542)$ is shown in Figure~\ref{fig:31542-double-orth}.
\begin{figure}[ht]
\centering
\includegraphics[scale=0.65]{figures/31542-double-orth}
\caption{The double orthodontic sequence for $w = 31542$.}
\label{fig:31542-double-orth}
\end{figure}
\end{example}

For a polynomial $f\in \CC[x_1, \dots, x_n, y_1, \dots, y_n]$, define the operators
\begin{equation}
\label{eqn:pi-ij-operators}
\pi_{i,j} \colonequals \del_i((x_i + y_j)f), \qquad \overline\pi_{i,j}\colonequals \overline\del_i((x_i + y_j - x_iy_j)f).
\end{equation}
For $i\in[n]$ and $M\subseteq[n]$, define the quantities
\begin{equation}
\label{eqn:omega-iM-operators}
\omega_i^M\colonequals \prod_{\substack{j\in[i]\\m\in M}}(x_j + y_m),\qquad\overline\omega_i^M\colonequals \prod_{\substack{j\in [i]\\m\in M}} (x_j + y_m - x_jy_m).
\end{equation}
\begin{defi}Let $D$ be a \%-avoiding diagram with double orthodontic sequence $\mathbf K, \mathbf i, \mathbf j, \mathbf M$. Define
\[
\mathscr G_D(\mathbf x;\mathbf y)\colonequals \overline\omega_1^{K_1}\overline\omega_2^{K_2}\dots\overline\omega_n^{K_n}\overline\pi_{i_1,j_1}(\omega_{i_1}^{M_1}\overline\pi_{i_2,j_2}(\omega_{i_2}^{M_2}\dots\overline\pi_{i_\ell,j_\ell}(\omega_{i_\ell}^{M_\ell})\dots)),
\]
and write
\[
\mathscr S_D(\mathbf x;\mathbf y)\colonequals \omega_1^{K_1}\omega_2^{K_2}\dots\omega_n^{K_n}\pi_{i_1,j_1}(\omega_{i_1}^{M_1}\pi_{i_2,j_2}(\omega_{i_2}^{M_2}\dots\pi_{i_\ell,j_\ell}(\omega_{i_\ell}^{M_\ell})\dots)).\qedhere
\]
\end{defi}
\begin{rema}
\label{rem:sd-nonempty}
It can be shown by inducting down the orthodontic sequence that the monomial $\prod_{(i,j)\in D}x_i$ appears in $\mathscr S_D$ with coefficient $1$. As $\mathscr S_D$ is nonzero, it follows that $\mathscr S_D$ is the lowest degree part of $\mathscr G_D$.
\end{rema}
\begin{rema}
The polynomial $\mathscr G_D(\mathbf x; \mathbf y)$ is not invariant under permuting columns, in contrast to previous work~\cite{magyar98, mss22} on the orthodontia algorithm. This is by design: the Rothe diagram $D(2413)$ can be obtained from the Rothe diagram $D(132)$ by permuting the columns and adding a column equal to $\{1,2\}$, and although the ordinary Grothendieck polynomials satisfy $\mathfrak G_{2413}(\mathbf x) = x_1x_2 \mathfrak G_{132}(\mathbf x)$, there does not exist a polynomial $g(\mathbf x; \mathbf y)$ so that $\mathfrak G_{2413}(\mathbf x; \mathbf y) = g(\mathbf x; \mathbf y) \cdot \mathfrak G_{132}(\mathbf x; \mathbf y)$.
\end{rema}

\subsection{Orthodontic sort order on permutations}
Following~\cite{mss22}, we recall some basic facts about sorted permutations and the sorting projection.

In what follows, a \emph{standard interval} is a set of the form $[j]$ for some $j\geq 0$. Recall that a permutation $w \in S_n$ is called \emph{dominant} if it is $132$-avoiding. A permutation $w$ is dominant if and only if its Rothe diagram $D(w) = (D(w)_1, \dots, D(w)_n)$ consists only of standard intervals, which necessarily satisfy $\#D(w)_1 \geq \dots \geq \#D(w)_n$.
\begin{defi}
Fix a permutation $w \in S_n$. The \emph{primary column data} of $w$, denoted $(h,C,\alpha,i_1,\beta)$, is defined as follows.

If $w$ is not dominant, the diagram $D(w)$ has a column which is not a standard interval. Set $h$ to be the smallest integer such that the column $D(w)_{h+1}$ is not a standard interval. Set $C$ to be the column $D(w)_{h+1}\subseteq [n]$. Set $\alpha$ to be the largest integer such that $[\alpha]\subseteq C$. Set $i_1$ to be the smallest missing tooth of $C$. Lastly, set $\beta=i_1 - \alpha$, the size of the ``uppermost gap'' of $C$.

If $w$ is dominant, set $h=n$, $C=\emptyset$, $\alpha=0$, $i_1=n$, and $\beta=n$. 
\end{defi}
\begin{lemm}[{\cite[Lem 3.4]{mss22}}]
\label{lem:sigma-sort}
Any permutation $w$ restricts to a bijection $[\alpha+1, i_1]\to [h-\beta+1,h]$. The corresponding permutation $\sigma\in S_\beta$ is dominant.
\end{lemm}
See Figure~\ref{fig:D68342751-and-sort} for an example.

For $w\in S_n$, write $\sigma(w)$ for the dominant permutation obtained by restricting $w$ to $[\alpha+1, i_1]$.
\begin{defi}
The permutation $w$ is \emph{sorted} if $\sigma(w)$ is the identity. The \emph{sorting} of $w$, denoted $w_\sort$, is the permutation obtained from $w$ by reordering $w(\alpha+1), \dots, w(i_1)$ to be in increasing order.
\end{defi}

\begin{example}
\label{ex:running-1}
Let $w = 68342751$. The primary column data of $w$ is 
\[h=4,\,\,C=\{1,2,6\},\,\,\alpha=2,\,\,i_1=5,\,\,\beta=3. \] 
Note that $w$ restricts to a bijection $ \{3,4,5\}\to\{2,3,4\}$; the corresponding permutation $\sigma(w) \in S_3$ is $\sigma(w) = 231$. The sorting of $w$ is $w_\sort = 68234751$. The Rothe diagrams of $w$ and $w_\sort$ are displayed in Figure~\ref{fig:D68342751-and-sort}.
\begin{figure}[ht]
\centering
\includegraphics{figures/D68342751-and-sort}
\caption{The Rothe diagram of $w = 68342751$ on the left, and of its sorting $w_\sort = 68234751$ on the right. In this example, $\sigma(w) = 231$.}
\label{fig:D68342751-and-sort}
\end{figure}
\end{example}

\begin{defi}
\label{defn:os}
The \emph{orthodontic sort order} $\leq_\os$ is the reflexive and transitive closure of the relations
\begin{align*}
\label{eqn:os1}
w_{\textup{sort}}&\preceq w \tag{$\dagger$}\\
\label{eqn:os2} w s_{i_1} \dots s_{\alpha + 1}&\preceq w\textup{ whenever $w$ is nonidentity and sorted}. \tag{$\ddagger$}
\end{align*}
\end{defi}
\begin{prop}[{\cite[Prop 5.6]{mss22}}]
The relation $\leq_\os$ is a partial order on $S_n$ and the identity permutation is the minimum element.
\end{prop}
\section{Orthodontia and double Grothendieck polynomials}
We establish Theorem~\ref{thm:double-groth-master} by studying the relationship between $\mathfrak G_w$ and $\mathfrak G_{w_\sort}$ (Proposition~\ref{prop:os1-groth}), the relationship between $\mathscr G_{D(w)}$ and $\mathscr G_{D(ws_{i_1}\dots s_\alpha)}$ (Theorem~\ref{thm:os2-orth}), and then inducting on the orthodontic sort order (Definition~\ref{defn:os}). 

The following example motivates Proposition~\ref{prop:rothe-of-sort} and Corollary~\ref{cor:os1-orth}.
\begin{example}
\label{ex:running-2}
Let $w = 68342751$, as in Example~\ref{ex:running-1}, so that $w_\sort = 68234751$. Observe that $D(w_\sort) = D(w) \setminus \{(2,3), (2,4)\}$ is obtained from $D(w)$ by removing bottom-aligned portions of standard interval columns. Proposition~\ref{prop:rothe-of-sort} asserts that this phenomenon holds in general.

Write $\mathbf K(w) = (K_1, \dots, K_n)$ and $\mathbf K(w_\sort) = (K_1', \dots, K_n')$. Then:
\[
K_1 = \emptyset, \quad K_2 = \{3,4\}, \quad K_3 = \emptyset, \quad K_4 = \{2\}, \quad K_5 = \emptyset, \quad K_6 = \emptyset, \quad K_7 = \{1\}
\]
and
\[
K_1' = \emptyset, \quad K_2' = \{2,3,4\}, \quad K_3' = \emptyset, \quad K_4' = \emptyset, \quad K_5' = \emptyset, \quad K_6' = \emptyset, \quad K_7' = \{1\}.
\]
The fact that $K_4' = K_4 \setminus \{2\}$ and $K_2' = K_2 \cup \{2\}$ can be viewed as a consequence of the fact that $D(\sigma(w)) = D(231)$ has Rothe diagram consisting of one standard interval column of size two.

As $D(w_\sort)$ can be obtained from $D(w)$ by removing bottom-aligned portions of standard interval columns, the two diagrams agree after removing standard interval columns. It follows that the $\mathbf i,\mathbf j, \mathbf M$ orthodontic sequences of $w$ and $w_\sort$ agree: $\mathbf i(w) = \mathbf i(w_\sort)$, $\mathbf j(w) = \mathbf j(w_\sort)$, and $\mathbf M(w) = \mathbf M(w_\sort)$. This phenomenon holds more generally, and is stated precisely in Corollary~\ref{cor:os1-orth}.
\end{example}

\begin{prop}
\label{prop:rothe-of-sort}
Let $(h,C,\alpha,i_1,\beta)$ denote the primary column data of $w\in S_n$. Set $\lambda^\sigma_j \colonequals \#D(\sigma(w))_j$. The Rothe diagram $D(w_\sort)$ can be obtained from $D(w)$ by removing bottom-aligned portions of standard-interval columns, and
\begin{equation}
\label{eqn:rothe-of-sort}
D(w) \setminus D(w_\sort) = \{(\alpha+a,h-\beta+b)\colon 1 \leq a\leq\lambda^\sigma_b \textup{ and } 1\leq b\leq \beta\}.
\end{equation}
\end{prop}

\begin{proof}[Proof of Proposition~\ref{prop:rothe-of-sort}]
As $w_\sort$ is obtained from $w$ by reordering numbers $\{w^{-1}(j)\colon j \in [h-\beta+1,h]\}$, the columns $D(w_\sort)_j$ and $D(w)_j$ agree whenever $j\not\in [h-\beta+1,h]$. By definition of the primary column data and of the sorting of a permutation, $D(w)_{h-\beta+b} = [\alpha+\lambda^\sigma_b]$ and $D(w_\sort)_{h-\beta+b} = [\alpha]$.
\end{proof}
\begin{coro}[cf.~{\cite[Prop 4.4]{mss22}}]
\label{cor:os1-orth}
Let $(h,C,\alpha,i_1,\beta)$ denote the primary column data of $w\in S_n$. Set 
\[
\mathcal S_a \colonequals \left\{h-\beta+j\colon\begin{array}{c} j\in [\beta]\\ \#D(\sigma(w))_j = a\end{array}\right\}.
\]
Write $\mathbf K(w) = (K_1, \dots, K_n)$ and $\mathbf K(w_\sort) = (K_1', \dots, K_n')$. Then
\[
\mathbf i(w) = \mathbf i(w_\sort), \qquad  \mathbf j(w) =  \mathbf j(w_\sort), \qquad  \mathbf M(w) = \mathbf M(w_\sort),
\]
and
\[
K_j = \begin{cases} K_j' &\textup{ if } j\leq \alpha-1,\\ K_j' \setminus \{h-\beta+1, \dots, h \} \cup \mathcal S_0 &\textup{ if } j = \alpha,\\ K_j' \cup \mathcal S_a &\textup{ if } j = \alpha + a, \textup{ where } a\in[\beta],\\
K_j' &\textup{ if } j \geq i_1 + 1.  \end{cases}
\]
In particular,
\[
\mathscr G_{D(w)} = \left(\prod_{(a,b)\in D(w)\setminus D(w_\sort)} (x_a + y_b - x_ay_b)\right) \cdot\mathscr G_{D(w_\sort)}.
\]
\end{coro}
\begin{proof}
Proposition~\ref{prop:rothe-of-sort} implies that the diagrams $D(w)$ and $D(w_\sort)$ agree after removing their standard interval columns. It follows that the orthodontic sequences $\mathbf i, \mathbf j, \mathbf M$ of $w$ and $w_\sort$ agree. The formulas for $K_j$ in terms of $K_j'$ hold by~\eqref{eqn:rothe-of-sort}. The relationship between $\mathscr G_{D(w)}$ and $\mathscr G_{D(w_\sort)}$ follows from the construction~\eqref{eqn:double-groth-master} of $\mathscr G_D$.
\end{proof}
\begin{lemm}
\label{lem:CwEw}
Let $(h,C,\alpha,i_1,\beta)$ denote the primary column data of $w\in S_n$. For $j\in[h]$, set 
\[\lambda_j\colonequals \#D(w)_j \quad\mbox{and}\quad \nu_j\colonequals \#\{i\leq j\colon \#D(w)_i \geq w^{-1}(j)\}.\]
Fix any $P \in \PD(w)$. Then for every $j\in[h]$, the pipe $P^{(j)}$ begins by traversing $\lambda_j$ crossing tiles vertically, ends by traversing $\nu_j$ tiles horizontally, and otherwise traverses only bump tiles.
In particular, every tile in
\[
C_w \colonequals \left\{(i,j)\colon \begin{array}{c}1 \leq i \leq \lambda_j,\\ 1\leq j \leq h\end{array}\right\}
\]
is a cross, and every tile in
\[
E_w \colonequals \left\{(i,j)\colon \begin{array}{c} h-\beta+1 \leq i \leq h,\\\alpha+1 \leq j \leq i_1,\\i + j \leq h + \alpha + 1\end{array}\right\} \setminus C_w
\]
is an elbow.
\end{lemm}
\begin{example}
Figure~\ref{fig:typical-pd} gives an example of a typical pipe dream, with the structure guaranteed by Lemma~\ref{lem:CwEw} highlighted.
\begin{figure}[ht]
\centering
\includegraphics{figures/typical-pd}
\caption{Left: The Rothe diagram of a permutation $w = abc3245d16...$, for $a,b,c,d > 6$. Right: A typical pipe dream $P\in\PD(w)$. The boxes in $C_w$ are shaded purple and the boxes in $E_w$ are shaded red.}
\label{fig:typical-pd}
\end{figure}
\end{example}
\begin{proof}[Proof of Lemma~\ref{lem:CwEw}]
Induct on $j$. The pipe $P^{(1)}$ necessarily traverses $\lambda_1$ cross tiles vertically before traversing a bump tile to exit at the $w^{-1}(1)$\textsuperscript{th} row. Now assume that, for all $k < j$, the pipes $P^{(k)}$ traverse $\lambda_k$ cross tiles vertically, then traverse bump tiles, before ending by traversing $\nu_k$ tiles horizontally.

As $\lambda_1 \geq \dots \geq \lambda_j$, the inductive hypothesis guarantees that all tiles
\[
\left\{(i,k)\colon \begin{array}{c} 1 \leq i \leq \lambda_j \\ 1 \leq k \leq j-1\end{array}\right\}
\]
are cross tiles. Thus if $P^{(j)}$ traverses a bump tile after traversing $a < \lambda_j$ cross tiles vertically, pipe $P^{(j)}$ is forced to travel horizontally  until exiting at the $(a+1)$\textsuperscript{th} row. As $a + 1 \leq \lambda_j < w^{-1}(j)$, this is a contradiction.

As $\lambda_i$ is weakly decreasing, $\#D(w)_i \geq w^{-1}(j)$ if and only if $i \in [\nu_j]$. The inductive hypothesis guarantees that the pipe exiting at row $w^{-1}(j)$ traverses $\nu_j$ crossing tiles horizontally before traversing any bump tiles.

It remains to show that $P^{(j)}$ does not traverse any other cross tiles. To this end, observe that any such cross tile $(a,b)$ satisfies $a > \lambda_j$ and $b > \nu_j$. It follows that any such cross tile cannot involve $P^{(w^{-1}(k))}$ for $k \leq \lambda_j$ as the paths of these pipes are contained in $\{(a,b)\colon a \leq k\}$, and cannot involve $P^{(k)}$ for $k \leq \nu_j$ as the paths of these pipes are contained in $\{(a,b)\colon b \leq \nu_j\}$. In other words, any such cross tile involves $P^{(j)}$ and $P^{(k)}$ for $k\not\in w^{-1}([\lambda_j])$ and $k\not\in[\nu_j]$.

Then, the first part of the result follows from the fact that the inversions of $w$ involving $j$ are
\[
\{(k,j)\colon k \in w^{-1}([\lambda_j]) \textup{ or } k \in[\nu_j]\}.
\]
The rest of the lemma follows from the fact that the set of pipes entering the top edge of the triangular grid
\[
\left\{(i,j)\colon \begin{array}{c} h-\beta+1 \leq i \leq h,\\\alpha+1 \leq j \leq i_1,\\i + j \leq h + \alpha + 1\end{array}\right\},
\]
namely the set $\{P^{(j)}\colon h-\beta+1\leq j\leq h\}$, is equal to the set of pipes exiting the left edge, hence restricts to a pipe dream for $\sigma(w)$.
\end{proof}

\begin{coro}
\label{cor:crossings}
Let $(h,C,\alpha,i_1,\beta)$ denote the primary column data of $w\in S_n$, and let $\mathcal S\colonequals \{h-\beta+1, \dots, h\}$. Fix $P \in \PD(w)$.
\begin{itemize}
\item For $i,j\in\mathcal S$ and $k\not\in\mathcal S$, the pipes $P^{(i)}$ and $P^{(k)}$ cross if and only if $P^{(j)}$ and $P^{(k)}$ cross.
\item For $i\in\mathcal S$ and $j\in[n]$, the pipes $P^{(i)}$ and $P^{(j)}$ cross at most once.
\item For $i,j\in\mathcal S$, the pipes $P^{(i)}$ and $P^{(j)}$ can cross only at $C_w \setminus C_{w_\sort}$, and conversely every tile in $C_w \setminus C_{w_\sort}$ is a crossing of pipes $P^{(i)}$ and $P^{(j)}$ for $i,j\in\mathcal S$.
\end{itemize}
\end{coro}
\begin{proof}
For any $i\in\mathcal S$, Lemma~\ref{lem:CwEw} guarantees that pipe $P^{(i)}$ begins by traversing $\alpha$ many cross tiles vertically, crossing pipes $P^{(k)}$ for $k \in w^{-1}([\alpha])$, and ends by traversing $h-\beta$ many cross tiles horizontally, crossing pipes $P^{(k)}$ for $k\in [h-\beta]$. Lemma~\ref{lem:CwEw} also guarantees that the tiles in $C_w \cup E_w \setminus C_{w_\sort}$ involve only the pipes $P^{(i)}$ for $i\in\mathcal S$. All three parts of the claim follow.

\end{proof}
\begin{prop}[cf.~{\cite[Prop 3.9]{mss22}}]
\label{prop:os1-groth}
Let $w\in S_n$ and set $\lambda_i\colonequals \#D(\sigma(w))_i$. Then
\[
\mathfrak G_w = \left(\prod_{(a,b) \in D(w)\setminus D(w_\sort)} (x_a + y_b - x_ay_b) \right)\cdot\mathfrak G_{w_{\sort}}.
\]
\end{prop}
\begin{proof}
Fix a pipe dream $P \in \PD(w)$. By Proposition~\ref{prop:rothe-of-sort} and Lemma~\ref{lem:CwEw}, the tiles in $D(w) \setminus D(w_\sort)$ are crosses. Let $F(P)$ be the pipe dream obtained from $P$ by replacing the crosses in $D(w)\setminus D(w_\sort)$ with elbows. By uncrossing all pipes $P^{(i)}$ and $P^{(j)}$ with $i,j\in\mathcal S$, Corollary~\ref{cor:crossings} guarantees that the Demazure word $\del(F(P))$ of $F(P)$ is $w_\sort$.

For any pipe dream $P\in\PD(w_\sort)$, let $G(P)$ be the pipe dream obtained from $P$ by replacing the elbows in $D(w)\setminus D(w_\sort)\subseteq E_{w_\sort}$ by crosses. Corollary~\ref{cor:crossings} guarantees that the Demazure word $\del(G(P))$ is $w$. As $FG \colon \PD(w_\sort)\to\PD(w_\sort)$ and $GF\colon \PD(w)\to \PD(w)$ are the identity, the map $F$ is a bijection.

The claim follows from the fact that $F\colon \PD(w)\to\PD(w_\sort)$ scales the weight of a pipe dream by the monomial
\[
\prod_{(a,b) \in D(w)\setminus D(w_\sort)} (x_a + y_b - x_ay_b).\qedhere
\]
\end{proof}

The following example motivates Theorem~\ref{thm:os2-orth}, which gives a precise description of the first few terms of the orthodontic sequences of sorted permutations.
\begin{example}[cf.~{\cite[Ex 4.1]{mss22}}]
Consider the sorted permutation $w = 68234751$, with Rothe diagram depicted in the left of Figure~\ref{fig:68234751-seq}. The Rothe diagram of $w' \colonequals ws_5s_4s_3$ ``almost'' appears in the double orthodontic sequence for $D(w)$, as shown in the top of Figure~\ref{fig:68234751-seq}. Part (6) of Theorem~\ref{thm:os2-orth} makes this connection precise.

\begin{figure}
\begin{center}
\includegraphics[scale=0.72]{figures/68234751-seq}
\end{center}
\caption{Left: Rothe diagram $D(w)$ for $w = 68234751$. Middle: first few orthodontic steps for $D(w)$. Top: Rothe diagram of $D(w')$ for $w' = ws_5s_4s_3 = 68723451$.}
\label{fig:68234751-seq}
\end{figure}

The primary column data $(h,C,\alpha,i_1,\beta)$ of $w$ is given by $h = 4$, $C = \{1,2,6\}$, $\alpha = 2$, $i_1 = 5$, and $\beta = 3$.
From that the uppermost gap has size $\beta = 3$ and the first missing tooth $i_1 = 5$ occurs in column $h + 1 = 5$, it follows the next missing teeth are $i_2 = i_1 - 1 = 4$ and $i_3 = i_2 - 1 = 3$ and which occur in the fifth column. This phenomenon is captured as part (1) of Theorem~\ref{thm:os2-orth}. Direct computation gives $j_1 = 5 - 3 = 2$, $j_2 = j_1 + 1 = 3$, and $j_3 = j_2 + 1 = 4$; this is part (2) of Theorem~\ref{thm:os2-orth}. None of these row-swap orthodontic moves introduce standard interval columns; this is part (5) of Theorem~\ref{thm:os2-orth}.

As $w$ is sorted, the three columns immediately to the left of column $h+1 = 5$ (i.e., columns $2,3,4$) are standard intervals equal to $[\alpha] = [2]$. This is part (3) of Theorem~\ref{thm:os2-orth}. Furthermore, there are no standard interval columns of size $k$ for any $k \in [\alpha+1, i_1] = \{3,4,5\}$. This is part (4) of Theorem~\ref{thm:os2-orth}.
\end{example}

\begin{thm}[cf.~{\cite[Thm 4.2]{mss22}}]
\label{thm:os2-orth}
Let $w\in S_n$ be a nonidentity sorted permutation, and suppose $w$ has orthodontic sequence 
\[
\mathbf i = (i_1, \dots, i_\ell), \qquad \mathbf j = (j_1, \dots, j_\ell), \qquad \mathbf K = (K_1, \dots, K_n),\qquad \mathbf M = (M_1, \dots, M_\ell).
\]
Let $(h,C,\alpha,i_1,\beta)$ be the primary column data of $w$. Write $\mathcal S\colonequals \{h-\beta+1,\dots,h\}$. Then:
\begin{enumerate}
\item For $k\in[\beta]$, we have $i_k = i_1 - k + 1$.
\item For $k\in[\beta]$, we have $j_k = h - \beta + k$.
\item If $\alpha > 0$, then $K_\alpha \supseteq \mathcal S$.
\item For $k\in[\alpha+1,i_1]$, we have $K_k = \emptyset$.
\item For $k\in[\beta-1]$, we have $M_k = \emptyset$. 
\item The permutation $w'\colonequals ws_{i_1}\dots s_{\alpha+1}$ has orthodontic sequence
\[
\mathbf i(w') = (i_{\beta+1}, \dots, i_\ell), \qquad \mathbf j(w') = (j_{\beta+1}, \dots, j_\ell), \qquad \mathbf M(w') = (M_{\beta+1},\dots,M_\ell),
\]
and
\[
\mathbf K(w') = \begin{cases} (K_1, \dots, K_{\alpha-1}, K_\alpha\setminus \mathcal S,\mathcal S\sqcup M_\beta, K_{\alpha+2}, \dots, K_n) &\textup{ if } \alpha > 0\\ (\mathcal S \sqcup M_\beta, K_2, \dots, K_n) &\textup{ if } \alpha = 0.\end{cases}
\]
In particular,
\begin{equation} \tag{$\star$} \label{eqn:Gw-vs-Gw'}
\mathscr G_{D(w)} = \overline\pi_{i_1,h+1-\beta} \dots \overline\pi_{\alpha+1,h}\left(\prod_{s\in \mathcal S} (x_{\alpha + 1} + y_s - x_{\alpha+1}y_s)^{-1} \cdot \mathscr G_{D(w')}\right).
\end{equation}
\end{enumerate}
\end{thm}
\begin{proof}
By definition, $C = D(w)_{h+1}$ contains $[\alpha]\cup\{i_1 + 1\}$ and does not contain any of $\alpha + 1, \dots, i_1$. Thus, the leftmost missing teeth of the diagrams in the orthodontic sequence of $D(w)$ are equal to $(i_1, i_1 - 1, \dots, \alpha+1)$, and all occur at column $h+1$, with corresponding sequence of uppermost gaps $(\beta, \beta - 1, \dots, 1)$. Parts (1) and (2) follow. 

As $w$ is sorted, the columns $D(w_\sort)_{h-\beta+b}$ are equal to $[\alpha]$, and part (3) follows. Parts (4) and (5) are proven as~\cite[Thm 4.2, (iii), (iv)]{mss22} respectively. 

In order to relate the orthodontic sequences of $w$ and $w'$, consider the diagrams
\begin{align*}
E &= (\underbrace{\emptyset, \dots, \emptyset}_{\textup{$h$ many}}, D(w)_{h+1}, \dots, D(w)_n),\\
E' &= (\underbrace{\emptyset, \dots, \emptyset}_{\textup{$h$ many}}, D(w')_{h+1}, \dots, D(w')_n).
\end{align*}
Note that $E' = E\cdot s_{i_1} \dots s_{\alpha+1}$ and that $M_\beta = \{j\colon E'_j = [\alpha+1]\}$. As the first $h$ columns of $D(w)$ and $D(w')$ are standard intervals, it follows that $D(w')_-$ occurs in the orthodontic sequence of diagrams for $D(w)$ and furthermore that
\[
\mathbf i(w') = (i_{\beta+1}, \dots, i_\ell), \qquad \mathbf j(w') = (j_{\beta+1}, \dots, j_\ell), \qquad \mathbf M(w') = (M_{\beta+1},\dots,M_\ell).
\]
From the facts that 
\begin{itemize}
	\item $D(w)_j = D(w')_j$ for $j \leq h-\beta$,
	\item $D(w)_j \cup \{\alpha+1\} = D(w')_j$ for $h-\beta+1\leq j\leq h$, and
	\item $M_\beta = \{j\colon E'_j = [\alpha+1]\}$,
\end{itemize}
the formula for $\mathbf K(w')$ in terms of $\mathbf K(w)$ follows. This gives part (6).

Finally, the equation~\eqref{eqn:Gw-vs-Gw'} is a consequence of part (4) of the claim via the string of equalities
\begin{align*}
\mathscr G_{D(w)} &= \overline\omega_1^{K_1}\dots\overline\omega_n^{K_n}\overline\pi_{i_1,h+1-\beta}\dots\overline\pi_{\alpha+1,h}(\mathscr G_{D(w')_-})
\\&\overset{(4)}=\overline\pi_{i_1,h+1-\beta}\dots\overline\pi_{\alpha+1,h}\left(\overline\omega_1^{K_1}\dots\overline\omega_n^{K_n}\mathscr G_{D(w')_-}\right)
\\&= \overline\pi_{i_1,h+1-\beta} \dots \overline\pi_{\alpha+1,h}\left(\prod_{s\in \mathcal S} (x_{\alpha + 1} + y_s - x_{\alpha+1}y_s)^{-1} \cdot \mathscr G_{D(w')}\right).\qedhere
\end{align*}
\end{proof}

In order to relate the formula from Theorem~\ref{thm:os2-orth} to Grothendieck polynomials, the following lemmas will be useful.

\begin{lemm}
\label{lem:operator-trick}
Let $\overline\Delta_a\colonequals\overline\del_{i+a}\circ\dots\circ \overline\del_i$. For $a = -1$, set $\overline\Delta_a \colonequals 1$ to be the identity operator. Given $g\in\CC[\mathbf x,\mathbf y]$ and $\ell \geq 1$ such that $\del_{i+\ell}(g) = 0$, the equality
\[
\overline\pi_{i+\ell, j_\ell}\circ \overline\Delta_{\ell-1}(g) = \overline\del_{i+\ell}\circ \overline\pi_{i+\ell-1, j_\ell}\circ \overline\Delta_{\ell-2}(g)
\]
holds. In particular, for any $\ell \geq 0$,
\[
\overline\pi_{i+\ell, j_\ell}\circ \overline\Delta_{\ell-1}(g) = \overline\Delta_\ell((x_i + y_{j_\ell} - x_i y_{j_\ell})g)
\]
whenever $\del_{i+1}(g) = \dots = \del_{i+\ell}(g) = 0$.
\end{lemm}
\begin{proof}
Work in the ring of endomorphisms of $\CC[\mathbf x, \mathbf y]$. For $h\in\CC[\mathbf x, \mathbf y]$, write $m_h$ for the ``multiplication by $h$'' operator, given by $f\mapsto fh$. As
\[
\del_i \circ m_{x_i} = (m_{x_{i+1}}\circ \del_i) + 1 \qquad\textup{ and } \qquad \del_i\circ m_{y_j} = m_{y_j}\circ \del_i\textup{ for all $j$},
\]
observe that
\begin{align*}
m_{x_{i+1} - y_j + x_{i+1}y_j}\circ \del_i &= (m_{x_{i+1}}\circ m_{1+y_j} - m_{y_j}) \circ \del_i \\&= (\del_i\circ m_{x_i} - 1)\circ m_{1+y_j} - \del_i\circ m_{y_j}\\&\label{eqn:commute}= \del_i\circ (m_{x_i - y_j + x_iy_j}) - m_{1+y_j}.\tag{$\diamondsuit$}
\end{align*}
Hence
\begin{align*}
\overline\pi_{i+\ell, j_\ell}\circ \overline\Delta_{\ell-1} &= \overline\del_{i+\ell}\circ m_{x_{i+\ell} + y_{j_\ell} - x_{i+\ell}y_{j_\ell}}\circ \del_{i+\ell-1}\circ m_{1 - x_{i+\ell}} \circ \overline\Delta_{\ell-2}\\
&\overset{\eqref{eqn:commute}}=\overline\del_{i+\ell} \circ (\del_{i+\ell-1} \circ m_{x_{i+\ell-1} + y_{j_\ell} - x_{i+\ell-1}y_{j_\ell}} - m_{1+y_{j_\ell}}) \circ m_{1-x_{i+\ell}}\circ\overline\Delta_{\ell-2}\\
\label{eqn:operator-trick}
&=\overline\del_{i+\ell}\circ\overline\pi_{i+\ell-1,j_\ell}\circ\overline\Delta_{\ell-2} - \overline\del_{i+\ell}\circ m_{1+y_{j_\ell}}\circ m_{1-x_{i+\ell}} \circ\overline\Delta_{\ell-2}.\tag{$*$}
\end{align*}
Note that $\overline\del_{i+\ell}$ and $m_{1-x_{i+\ell}}$ commute with $\overline\Delta_{\ell-2}$, so that
\begin{align*}
\overline\del_{i+\ell}\circ m_{1+y_{j_\ell}}\circ m_{1-x_{i+\ell}} \circ\overline\Delta_{\ell-2}(g) &= m_{1+y_{j_\ell}}\circ \overline\Delta_{\ell-2}\circ\overline\del_{i+\ell}\circ m_{1-x_{i+\ell}} (g)
\\&=\label{eqn:vanishing-trick}m_{1+y_{j_\ell}}\circ \overline\Delta_{\ell-2}\circ\underbrace{\del_{i+\ell}((1-x_{i+\ell+1})(1-x_{i+\ell})g)}_{=0}.\tag{$**$}
\end{align*}
Applying the equality~\eqref{eqn:operator-trick} of operators to the function $g$, and using~\eqref{eqn:vanishing-trick}, the claimed equality
\[
\overline\pi_{i+\ell, j_\ell}\circ \overline\Delta_{\ell-1}(g) = \overline\del_{i+\ell}\circ \overline\pi_{i+\ell-1, j_\ell}\circ \overline\Delta_{\ell-2}(g)
\]
follows.

The second part of the lemma, for $\ell = 0$, is nothing but the identity
\begin{equation}
\label{eqn:ell0-case}
\overline\pi_{i,j_0}(g) = \overline\del_i((x_i - y_{j_0})g),
\end{equation}
and in general it follows from repeated application of the first part of the lemma combined with the identity~\eqref{eqn:ell0-case}.
\end{proof}

\begin{lemm}[cf.~{\cite[Lem 5.11]{mss22}}]
\label{lem:pi-to-del}
Let $g$ be a polynomial with
\[
\del_{i+1}(g) = \dots = \del_{i+k}(g) = 0.
\]
Then
\[
\overline\pi_{i+k,j_k} \overline\pi_{i+k-1,j_{k-1}} \dots \overline\pi_{i,j_0}(g) = \overline\del_{i+k}\dots\overline\del_i\left(\prod_{a=0}^k(x_i + y_{j_a} - x_iy_{j_a}) \cdot g\right).
\]
\end{lemm}
\begin{proof}
Induct on $k$ as follows. The base case $k = 0$ is the identity~\eqref{eqn:ell0-case}. For $k > 0$, the inductive hypothesis guarantees that
\[
\pi_{i+k-1,j_{k-1}} \dots \overline\pi_{i,j_0}(g) = \overline\Delta_{k-1}\left(\prod_{a=0}^{k-1}(x_i + y_{j_a} - x_iy_{j_a}) \cdot g\right).
\]
Then, because
\[
\del_{i+1}\left(\prod_{a=0}^{k-1}(x_i + y_{j_a} - x_iy_{j_a})g\right) = \dots = \del_{i+k}\left(\prod_{a=0}^{k-1}(x_i + y_{j_a} - x_iy_{j_a})g\right) = 0,
\]
Lemma~\ref{lem:operator-trick} implies that
\[
\overline\pi_{i+k,j_k} \circ \overline\Delta_{k-1}\left(\prod_{a=0}^{k-1}(x_i + y_{j_a} - x_iy_{j_a})\cdot g\right) =\overline\Delta_k\left(\prod_{a=0}^k(x_i + y_{j_a} - x_iy_{j_a})\cdot g\right).\qedhere
\]
\end{proof}

\newtheorem*{thm:double-groth-master}{Theorem~\ref{thm:double-groth-master}}
\begin{thm:double-groth-master}
Let $D$ be a \%-avoiding diagram with double orthodontic sequence $\mathbf K, \mathbf i, \mathbf j, \mathbf M$. Define
\[
\mathscr G_D(\mathbf x;\mathbf y)\colonequals \overline\omega_1^{K_1}\overline\omega_2^{K_2}\dots\overline\omega_n^{K_n}\overline\pi_{i_1,j_1}(\overline\omega_{i_1}^{M_1}\overline\pi_{i_2,j_2}(\overline\omega_{i_2}^{M_2}\dots\overline\pi_{i_\ell,j_\ell}(\overline\omega_{i_\ell}^{M_\ell})\dots)).\tag{\ref{eqn:double-groth-master}}
\]
When $D = D(w)$ is the Rothe diagram of a permutation, then $\mathscr G_D(\mathbf x; \mathbf y) = \mathfrak G_w(\mathbf x; \mathbf y)$.
\end{thm:double-groth-master}
\begin{proof}[Proof of Theorem~\ref{thm:double-groth-master}]
Induct on the orthodontic sort order (Definition~\ref{defn:os}). In the base case $w = \mathrm{id}$, both $\mathscr G_{D(\mathrm{id})} = \mathscr G_{\emptyset}$ and $\mathfrak G_{\mathrm{id}}$ are equal to $1$.

Assume that $w$ is not sorted and that $\mathscr G_{D(w_\sort)} = \mathfrak G_{w_\sort}$. Corollary~\ref{cor:os1-orth} and Proposition~\ref{prop:os1-groth} imply that
\begin{align*}
\mathscr G_{D(w)} &= \left(\prod_{(a,b)\in D(w)\setminus D(w_\sort)}(x_a + y_b - x_ay_b)\right)\mathscr G_{D(w_\sort)}\\&=\left(\prod_{(a,b)\in D(w)\setminus D(w_\sort)}(x_a + y_b - x_ay_b)\right)\mathfrak G_{w_\sort}\\&=\mathfrak G_w.
\end{align*}
Now assume that $w$ is sorted and that $\mathscr G_{D(ws_{i_1}\dots s_\alpha)} = \mathfrak G_{ws_{i_1}\dots s_\alpha}$. Writing $w'\colonequals ws_{i_1}\dots s_\alpha$ and $\mathcal S\colonequals \{h-\beta+1, \dots, h\}$, Theorem~\ref{thm:os2-orth} and Lemma~\ref{lem:pi-to-del} imply that
\begin{align*}
\mathscr G_{D(w)} &= \overline\pi_{i_1,h+1-\beta} \dots \overline\pi_{\alpha+1,h}\left(\prod_{s\in \mathcal S} (x_{\alpha + 1} + y_s - x_{\alpha+1}y_s)^{-1} \cdot \mathscr G_{D(w')}\right)
\\&= \overline\del_{i_1}\dots\overline\del_\alpha\left(\mathscr G_{D(w')}\right)
\\&= \overline\del_{i_1}\dots\overline\del_\alpha\left(\mathfrak G_{w'}\right)
\\&= \mathfrak G_w.\qedhere
\end{align*}
\end{proof}
\begin{coro}
\label{cor:double-schub-master}
If $D = D(w)$ is a Rothe diagram, then $\mathscr S_D(\mathbf x; \mathbf y) = \mathfrak S_w(\mathbf x; -\mathbf y)$.
\end{coro}
\begin{proof}
By definition, $\mathfrak S_w(\mathbf x; \mathbf y)$ is the lowest degree part of $\mathfrak G_w(\mathbf x; -\mathbf y)$. Remark~\ref{rem:sd-nonempty} guarantees that $\mathscr S_D(\mathbf x; \mathbf y)$ is the lowest degree part of $\mathscr G_D(\mathbf x; \mathbf y)$. The result now follows from Theorem~\ref{thm:double-groth-master}.
\end{proof}

\section{Lascoux positivity}
We prove Theorem~\ref{thm:lascoux-positivity-main} by reducing the problem to the special case where every column is a standard interval (cf.~Lemma~\ref{lem:overlinepi-positivity} and Corollary~\ref{cor:omegas-then-pis}). This case can be checked explicitly (Proposition~\ref{prop:base-positivity}).

Recall the operator $r_{m,n}$ introduced in~\cite{yu23} that acts on $n$ variable polynomial via
\[
f(x_1, \dots, x_n)\mapsto x_1^m\dots x_n^mf(x_n^{-1},\dots, x_1^{-1}).
\]

\begin{lemm}
For $i\in[n]$ and $M\subseteq[n]$, and $f\in\CC[\mathbf x]$ satisfying $\deg_{x_i}(f) \leq m$, the equalities
\begin{align}
\label{eqn:omega-spec}
\omega_i^M |_{y_j \mapsto -1} &= (x_1 - 1)^{|M|}\dots(x_i - 1)^{|M|}\\
\label{eqn:omega-int}
r_{m+1,n}\left((x_1 - 1)\dots(x_i - 1)f\right) &=x_1\dots x_{n-i}(1 - x_{n-i+1})\dots(1 - x_n) r_{m,n}(f)\\
\label{eqn:pi-spec}
\pi_{i,j}(f)|_{y_j\mapsto -1} &= \del_i((x_i- 1)(f|_{y_j\mapsto1}))\\
\label{eqn:pi-int}
r_{m,n}(\del_i((x_i- 1)f)) &= \overline\pi_{n-i}(r_{m,n}(f))
\end{align}
hold.
\end{lemm}
\begin{proof}
Equations~\eqref{eqn:omega-spec} and~\eqref{eqn:pi-spec} are immediate from the definition of $\omega_i^M$ and $\pi_{i,j}$. Equations~\eqref{eqn:omega-int} and~\eqref{eqn:pi-int} can be proven by a manual check when $f$ is a monomial and applying linearity. (Equation~\eqref{eqn:pi-int} also follows from~\cite[Lem 3.4]{yu23}).
\end{proof}

Let $\varphi_i$ denote the operator 
\[
\varphi_i\colon f\mapsto x_1 \dots x_i (1 - x_{i+1})\dots(1 - x_n)f.
\]
\begin{coro}
\label{cor:rmn-via-operators}
The polynomial $r_{m,n}(\mathscr S_D(\mathbf x; -\mathbf 1))$ can be obtained from the polynomial $1\in\CC[\mathbf x]$ by repeated application of operators of the form $f\mapsto \overline\pi_i(f)$ and $\varphi_i$.
\end{coro}
\begin{proof}
By the orthodontic formula~\eqref{eqn:double-groth-master}, along with Equations~\eqref{eqn:omega-spec} and~\eqref{eqn:pi-spec}, $\mathscr S_D(\mathbf x; -\mathbf 1)$ can be obtained from the polynomial $1\in\CC[\mathbf x]$ by repeated application of operators of the form
\[
f\mapsto \del_i(x_i - 1)f \qquad\textup{ and } \qquad f\mapsto (x_1 - 1)\dots (x_i - 1)f.
\]
Using Equations~\eqref{eqn:omega-int} and~\eqref{eqn:pi-int} to commute the operator $r_{m,n}$ past the two operators above, it follows that $r_{m,n}(\mathscr S_D(\mathbf x; -\mathbf 1))$ can be obtained by repeated application of the operators $\overline\pi_i$ and $\varphi_{n-i}$.
\end{proof}

\begin{lemm}
\label{lem:overlinepi-positivity}
The equality
\[
\overline\pi_i(\mathfrak L_\alpha) = \begin{cases} \mathfrak L_\alpha &\textup{ if } \alpha_i > \alpha_{i+1}\\ \mathfrak L_{\alpha\cdot s_i} &\textup{ if } \alpha_i < \alpha_{i+1}\end{cases}
\]
holds; in particular, $\overline\pi_i$ preserves Lascoux positivity.
\end{lemm}
\begin{proof}
When $\alpha_i > \alpha_{i+1}$, the definition of Lascoux polynomials guarantees that $\mathfrak L_\alpha = \overline\pi_i\mathfrak L_{\alpha\cdot s_i}$. The claim then follows from the fact that $\overline\pi_i$ is idempotent. When $\alpha_i < \alpha_{i+1}$, the claim follows from the definition of $\mathfrak L_{\alpha \cdot s_i}$.
\end{proof}


\begin{prop}
\label{prop:mult-implies-main}
Conjecture~\ref{conj:mult-pos} implies Conjecture~\ref{conj:lascoux-positivity-general}.
\end{prop}
\begin{proof}
By Corollary~\ref{cor:rmn-via-operators}, it suffices to show that the operators $\overline\pi_i$ and $\varphi_i$ preserve Lascoux positivity. This follows from Lemma~\ref{lem:overlinepi-positivity} and Conjecture~\ref{conj:mult-pos}, respectively.
\end{proof}

\begin{lemm}
\label{lem:orth-of-incl}
Assume that the columns of $D$ are ordered by inclusion, and write $\mathbf K, \mathbf i, \mathbf j, \mathbf M$ for the orthodontic sequence. Then:
\begin{enumerate}
\item Every diagram appearing in the orthodontic sequence of $D$ also has all columns ordered by inclusion.
\item If $K_k\neq\emptyset$, then $i_j\neq k$ for all $j$.
\item If $M_k \neq \emptyset$ for some $k$, then $i_j \neq i_k$ for all $j > k$.
\end{enumerate}
\end{lemm}
\begin{proof}
Part 1 of the lemma follows from the fact that orthodontic moves either remove a column or swap two rows; both of these moves preserve the inclusion property of the columns.

If $K_k\neq\emptyset$, then $D$ has a column equal to $[k]$. As the columns of $D$ are ordered by inclusion, every other column of $D$ is either contained in $[k]$ or is equal to $[k]\sqcup S$ for some $S$; in particular, no empty box in the $k$\textsuperscript{th} row has a square below it in its column. As orthodontic moves remove columns or move boxes up, no empty box in the $k$\textsuperscript{th} row of any diagram $D'$ appearing in the orthodontic sequence of $D$ has a square below it in its column; in particular, $k$ can never be a missing tooth of $D'$. Part 2 of the lemma follows.

Similarly, if $M_k\neq\emptyset$ for some $k$, then there is a diagram $D'$ in the orthodontic sequence of $D$ which has a column equal to $[i_k]$. Arguing as above, it follows that $i_k$ can never be a missing tooth of a diagram in the orthodontic sequence of $D'$. Part 3 of the lemma follows. 
\end{proof}

\begin{coro}
\label{cor:omegas-then-pis}
Let $D$ be a diagram whose columns are ordered by inclusion, and let $\mathbf K, \mathbf i, \mathbf j, \mathbf M$ denote the orthodontic sequence of $D$. Then,
\[
\mathscr S_D(\mathbf x; \mathbf y) = \pi_{i_1,j_1}(\dots\pi_{i_\ell,j_\ell}(\omega_1^{K_1}\dots\omega_n^{K_n}\cdot \omega_{i_1}^{M_1}\dots\omega_{i_\ell}^{M_\ell})\dots).
\]
\end{coro}
\begin{proof}
Because $\omega_i^J$ is symmetric with respect to $x_1, \dots, x_i$ and also with respect to $x_{i+1}, \dots, x_n$, it follows that $\omega_i^J$ commutes with $\pi_{i',j}$ whenever $i' \neq i$. Lemma~\ref{lem:orth-of-incl} guarantees that the $\omega_i^J$ in the orthodontic formula~\eqref{eqn:double-groth-master} can be commuted past the $\pi_{i',j}$ occurring to the right. The claim follows.
\end{proof}
Let $C_{n,k,\ell}$ be the set of $\alpha = (\alpha_1, \dots, \alpha_n)\in\NN^n$ with $\alpha_1 = \dots = \alpha_k = \ell $ and $\alpha_j \leq \ell$ for all $j$.
\begin{lemm}
\label{lem:Cnkl}
Let $\alpha \in C_{n,k,\ell}$ and $i \leq k$. Then
\[
\mathfrak L_\alpha(x_1, \dots, x_n) = x_1^\ell \dots x_i^\ell \mathfrak L_{\alpha(i)}(x_{i+1},\dots, x_n), \qquad\textup{ where }\alpha(i)\colonequals (\alpha_{i+1}, \dots, \alpha_n).
\]
\end{lemm}
\begin{proof}
Write $\lambda\colonequals \sort(\alpha)$ for the partition obtained by reordering $\alpha$ so that the components are weakly decreasing. Then
\[
\mathfrak L_\alpha = \overline \pi_{i_1}\dots\overline \pi_{i_\ell}(\mathbf x^\lambda),
\]
where $i_j > k$ for all $j$. As $\lambda_1 = \dots = \lambda_k = \ell$, 
\[
\mathfrak L_\alpha = x_1^\ell\dots x_i^\ell  \overline \pi_{i_1}\dots\overline \pi_{i_\ell}(\mathbf x^{\lambda(i)}), \qquad \lambda(i)\colonequals (\lambda_{i+1}, \dots, \lambda_n)
\]
for any $i \leq k$. The claim follows from the fact that
\[
\overline \pi_{i_1}\dots\overline \pi_{i_\ell}(\mathbf x^{\lambda(i)}) = \mathfrak L_{\alpha(i)}(x_{i+1}, \dots, x_n).\qedhere
\]
\end{proof}
\begin{lemm}
\label{lem:Cnkl-basis}
The set of Lascoux polynomials $\{\mathfrak L_\alpha\colon \alpha \in C_{n,k,\ell}\}$ forms a basis for the vector space $V_{k,\ell}$ of polynomials of the form
\[
x_1^\ell \dots x_k^\ell \cdot f(x_{k+1}, \dots, x_n), \qquad \deg_{x_i}(f) \leq \ell.
\]
\end{lemm}
\begin{proof}
The set of polynomials $\mathfrak L_\alpha(x_{k+1}, \dots, x_n)$ with $\alpha_i\leq \ell$ forms a basis for the set of polynomials $\CC[x_{k+1}, \dots, x_n]$ such that $\deg_{x_i}(f) \leq \ell$ for all $\ell$. Lemma~\ref{lem:Cnkl} implies that $\{\mathfrak L_\alpha\colon \alpha \in C_{n,k,\ell}\}$ spans $V_{k,\ell}$. Since Lascoux polynomials are linearly independent, the claim follows.
\end{proof}

Recall that the stable Grothendieck polynomial $G_w(x_1, \dots, x_n)$ of a permutation $w\in S_m$ is defined to be the limit $\lim_{N\to\infty} \mathfrak G_{1^N\times w}(x_1, \dots, x_n, 0, 0,\dots)$.

\begin{example}
\label{ex:g21}
For $w = 21$, the stable Grothendieck polynomial $G_{21}(x_1, \dots, x_n)$ is equal to
\[
G_{21}(x_1, \dots, x_n) = \sum_{i=1}^n(-1)^{i+1}\mathbf e_i(\mathbf x),
\]
where $\mathbf e_i$ is the $i$\textsuperscript{th} elementary symmetric polynomial. (More generally, for permutations with a unique descent, $G_w(x_1, \dots, x_n)$ can be computed e.g.\ using~\cite[Thm~2.2]{lenart00}.)

In particular,
\[
(1-x_1)\dots(1-x_n) = 1 - G_{21}(x_1, \dots, x_n).\qedhere
\]
\end{example}

The key ingredient for the Lascoux positivity in Theorem~\ref{thm:lascoux-positivity-main} is the following result by Orelowitz and Yu.
\begin{thm}[{\cite[Thm 1.3]{oy23}}]
\label{thm:tianyi-hammer}
The product $\mathfrak L_\alpha(x_1, \dots, x_n) \cdot G_w(x_1, \dots, x_n)$ is a graded nonnegative sum of Lascoux polynomials:
\[
\mathfrak L_\alpha(x_1, \dots, x_n) \cdot G_w(x_1, \dots, x_n) = \sum_\beta c_\beta (-1)^{|\beta| - \ell(w) - |\alpha|} \mathfrak L_\beta(x_1, \dots, x_n), \qquad c_\beta \geq 0.
\]
\end{thm}
Orelowitz and Yu prove more in~\cite[Thm 1.3]{oy23}: they express the product as a sum over certain tableaux, each of which contribute a Lascoux polynomial in the expansion.

\begin{prop}
\label{prop:base-positivity}
Assume that $\varphi_i^{a_i}\varphi_{i+1}^{a_{i+1}}\dots\varphi_n^{a_n}$ is a graded nonnegative sum of Lascoux polynomials for some $a_i, a_{i+1}, \dots, a_n \geq 0$. Then
\[
\varphi_i \cdot \varphi_i^{a_i}\varphi_{i+1}^{a_{i+1}}\dots\varphi_n^{a_n}
\]
is again a graded nonnegative sum of Lascoux polynomials.
\end{prop}
\begin{proof}
For $i = n$, the result is trivial.

Assume that $i \leq n-1$ and write $f\colonequals \varphi_i^{a_i}\varphi_{i+1}^{a_{i+1}}\dots\varphi_n^{a_n}$. Write $\ell\colonequals a_i + \dots + a_n$ and observe that $f\in V_{i,\ell}$. Lemma~\ref{lem:Cnkl-basis} implies that the Lascoux expansion of $f$, graded nonnegative by assumption, reads
\[
f = \sum_{\alpha\in C_{n,i,\ell}}(-1)^{|\alpha| - \ell}c_\alpha\mathfrak L_\alpha(x_1, \dots, x_n), \qquad c_\alpha \geq 0.
\]
By Lemma~\ref{lem:Cnkl}, the polynomial $f$ expands as
\[
f = \sum_{\alpha \in C_{n,i,\ell}} (-1)^{|\alpha| - \ell} c_\alpha x_1^\ell \dots x_i^\ell \cdot \mathfrak L_{\alpha(i)}(x_{i+1}, \dots, x_n), \;\; \textup{ where } \alpha(i) \colonequals (\alpha_{i+1}, \dots, \alpha_n).
\]
As $\varphi_i = x_1 \dots x_i(1 - G_{21}(x_{i+1}, \dots, x_n))$ (cf.~Example~\ref{ex:g21}), the polynomial $\varphi_i f$ expands as
\begin{equation}
\label{eqn:varphiif-expansion}
\varphi_i f = \sum_{\alpha \in C_{n,i,\ell}}(-1)^{|\alpha| - \ell} c_\alpha x_1^{\ell+1}\dots x_i^{\ell+1} \cdot (1 - G_{21}(x_{i+1}, \dots, x_n)) \cdot \mathfrak L_{\alpha(i)}(x_{i+1}, \dots, x_n).
\end{equation}
Theorem~\ref{thm:tianyi-hammer} implies that
{\small \[
G_{21}(x_{i+1}, \dots, x_n) \mathfrak L_{\alpha(i)}(x_{i+1}, \dots, x_n) = \sum_{\beta(i)} c_{\alpha(i),\beta(i)} (-1)^{|\beta(i)| - 1 - |\alpha(i)|}\mathfrak L_{\beta(i)}(x_{i+1}, \dots, x_n).
\]}As $\deg_{x_j}(\mathfrak L_{\alpha(i)})\leq \ell$ and $\deg_{x_j}(G_{21}(x_{i+1}, \dots, x_n)) = 1$ for $i+1 \leq j \leq n$, it follows that $\deg_{x_j}(\mathfrak L_{\beta(i)}) \leq \ell+1$ for $i+1 \leq j\leq n$. Writing $\beta(i)\colonequals (\beta_{i+1}, \dots, n)$, Lemma~\ref{lem:Cnkl} implies that
\[
x_1^{\ell+1} \dots x_i^{\ell+1}\mathfrak L_{\beta(i)}(x_{i+1}, \dots, x_n) = \mathfrak L_\beta(x_1, \dots, x_n), \qquad \beta\colonequals (\underbrace{\ell, \dots, \ell}_{i \textup{ many}}, \beta_{i+1}, \dots, \beta_n).
\]
It follows that each summand in Equation~\eqref{eqn:varphiif-expansion} has a graded nonnegative Lascoux expansion.
\end{proof}

\newtheorem*{thm:lascoux-positivity-main}{Theorem~\ref{thm:lascoux-positivity-main}}
\begin{thm:lascoux-positivity-main}
Let $D\subseteq[n]\times[m]$ be a diagram whose columns are ordered by inclusion. Let $\mathscr S_D(\mathbf x; \mathbf y)$ be the lowest degree part of $\mathscr G_D(\mathbf x; \mathbf y)$. Then, the polynomial
\[
x_1^m \dots x_n^m \mathscr S_D(x_n^{-1}, \dots, x_1^{-1}; -1, \dots, -1)
\]
is a graded nonnegative sum of Lascoux polynomials $\mathfrak L_\alpha(x_1, \dots, x_n)$.
\end{thm:lascoux-positivity-main}
\begin{proof}
Corollary~\ref{cor:omegas-then-pis} asserts
\[
\mathscr S_D(\mathbf x; \mathbf y) = \pi_{i_1,j_1}(\dots\pi_{i_\ell,j_\ell}(\omega_1^{K_1}\dots\omega_n^{K_n}\cdot \omega_{i_1}^{M_1}\dots\omega_{i_\ell}^{M_\ell})\dots).
\]
Equations~\eqref{eqn:omega-spec}--\eqref{eqn:pi-int}, along with the fact that the $\varphi_i$ commute with each other, imply that
\[
r_{m,n}(\mathscr S_D(\mathbf x; -\mathbf 1)) = \overline\pi_{n - i_1}\dots\overline\pi_{n - i_\ell}(\varphi_{k_1}\dots\varphi_{k_m}(1))
\]
for $k_1 \geq \dots \geq k_m$. Proposition~\ref{prop:base-positivity} implies that $\varphi_{k_1}\dots\varphi_{k_m}(1)$ is Lascoux positive, and the result follows from Lemma~\ref{lem:overlinepi-positivity}.
\end{proof}
\newtheorem*{cor:lascoux-positivity-schub}{Corollary~\ref{cor:lascoux-positivity-schub}}

\begin{cor:lascoux-positivity-schub}
Let $w\in S_n$ be a vexillary permutation and denote the corresponding double Schubert polynomial by $\mathfrak S_w(x_1, \dots, x_n; y_1, \dots, y_n)$. Then the polynomial
\[
x_1^n \dots x_n^n \mathfrak S_w(x_n^{-1}, \dots, x_1^{-1}; 1, \dots, 1)
\]
is a graded nonnegative sum of Lascoux polynomials $\mathfrak L_\alpha(x_1, \dots, x_n)$.
\end{cor:lascoux-positivity-schub}
\begin{proof}
By Corollary~\ref{cor:double-schub-master}, the double Schubert polynomial $\mathfrak S_w(\mathbf x; \mathbf 1)$ is equal to $\mathscr S_{D(w)}(\mathbf x; -\mathbf 1)$. The result follows from Theorem~\ref{thm:lascoux-positivity-main}.
\end{proof}

\begin{example}
	Replacing $w \in S_n$ with $w(n+1) \in S_{n+1}$ amounts to replacing 
	\[f\colonequals x_1^n\dots x_n^n\mathfrak S_w(x_n^{-1}, \dots, x_1^{-1};\mathbf 1)\quad \mbox{with}\quad x_1^{n+1}x_2\dots x_n x_{n+1} f(x_2,\dots,x_{n+1}).\]
	This operation preserves Lascoux positivity by Lemma~\ref{lem:Cnkl}. For example, if $w = 321$, then
	\begin{align*}
		\mathfrak{S}_w(\mathbf{x};\mathbf{y})&=\left(x_1-y_1\right) \left(x_2-y_1\right) \left(x_1-y_2\right)\\
		&=
			 x_1^2 x_2
			-x_1^2 y_1
			-x_1 x_2 y_1
			-x_1 x_2 y_2
			+x_1 y_1^2
			+x_1 y_1 y_2
			+x_2 y_1 y_2
			-y_1^2 y_2,
	\end{align*}
	which gives
	\begin{align*}
		x_1^3x_2^3x_3^3 \mathfrak S_w(x_3^{-1},x_2^{-1}, x_1^{-1}; \mathbf{1})
		&=
		x_1^3 x_2^2 x_3
		-x_1^3 x_2^3 x_3
		-x_1^3 x_2^2 x_3^2
		-x_1^3 x_2^2 x_3^2\\
		&\qquad
		+x_1^3 x_2^3 x_3^2
		+x_1^3 x_2^3 x_3^2
		+x_1^3 x_2^2 x_3^3
		-x_1^3 x_2^3 x_3^3\\
		&=\left(\mathfrak L_{321}\right) - \left(2\mathfrak L_{322} + \mathfrak L_{331}\right) + \left(\mathfrak L_{323}+\mathfrak L_{332}\right).
	\end{align*}
	Taking $w = 3214$ gives
	\begin{align*}
		x_1^4x_2^4x_3^4 x_4^4 \mathfrak S_w(x_4^{-1},x_3^{-1}, x_2^{-1}, x_1^{-1}; \mathbf{1})
		&=
		x_1^4 x_2^4 x_3^3 x_4^2
		-x_1^4 x_2^4 x_3^4 x_4^2
		-x_1^4 x_2^4 x_3^3 x_4^3
		-x_1^4 x_2^4 x_3^3 x_4^3\\
		&
		+x_1^4 x_2^4 x_3^4 x_4^3
		+x_1^4 x_2^4 x_3^4 x_4^3
		+x_1^4 x_2^4 x_3^3 x_4^4
		-x_1^4 x_2^4 x_3^4 x_4^4\\	
		&=\left(\mathfrak L_{4432}\right) - \left(2\mathfrak L_{4433} + \mathfrak L_{4442}\right) + \left(\mathfrak L_{4434}+\mathfrak L_{4443}\right).	\qedhere
	\end{align*}
\end{example}

\longthanks{We thank David Anderson, Sergey Fomin, Peter Magyar, Karola M\'esz\'aros, Alexander Yong, and Tianyi Yu for many helpful conversations and useful comments which improved our exposition.}

\bibliographystyle{mersenne-plain-nobysame}
\bibliography{ALCO_Setiabrata_1636}
\end{document}
