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\newcommand{\NN}{\mathbb N}
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\newcommand{\RR}{\mathbb R}

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\newcommand{\supp}{\mathrm{supp}}
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%% define your title in the usual way
\title{Grothendieck polynomials of fireworks and inverse fireworks permutations}


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%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
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\author{Jack Chen-An Chou\thanks{Email: c.chou@ufl.edu}\addressmark{1}, Linus Setiabrata\thanks{Email: setia@mit.edu}\addressmark{2}, \and Tianyi Yu\thanks{Email: yu.tianyi@uqam.ca}\addressmark{3}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}University of Florida, Gainesville, FL 32611, USA\\ \addressmark{2}Massachusetts Institute of Technology, Cambridge, MA 02139, USA\\ \addressmark{3}Laboratoire d'Alg\`ebre, de Combinatoire et d'Informatique Math\'ematique, Universit\'e du Qu\'ebec \`a Montr\'eal, Montreal QC, Canada}

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\received{\today}

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\abstract{
Grothendieck polynomials were introduced by Lascoux and Schützenberger
to represent $K$-classes of structure sheaves of Schubert varieties.
Their supports and top-degree components have recently become active areas of research. Using pipe dreams, combinatorial objects that compute Grothendieck polynomials, we prove several results for the Grothendieck polynomials of two families of permutations. For fireworks permutations, we show that the support of the Grothendieck polynomial is ``as large as possible'' and is M-convex after homogenization. For inverse fireworks permutations, we show that every monomial of non-maximal degree appearing in the Grothendieck polynomial divides another monomial of degree exactly one higher. Our work also yields the first direct combinatorial formula for the top-degree components of Grothendieck polynomials.}


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

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\keywords{Pipe dreams, Grothendieck polynomial, Castelnuovo--Mumford polynomial, Newton polytope}

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\articlenumber{89}
\begin{document}

\maketitle
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\section{Introduction}

Lascoux and Schützenberger~\cite{ls82a,ls82b} introduced explicit representatives of the cohomology classes and $K$-classes of structure sheaves of Schubert varieties in flag varieties, called the \emph{Schubert polynomials} $\mathfrak S_w(\mathbf x)$ and \emph{Grothendieck polynomials} $\mathfrak G_w(\mathbf x)$, respectively. In general, $\mathfrak G_w(\mathbf x)$ is not homogeneous, and its lowest-degree component equals $\mathfrak S_w(\mathbf x)$.

Recently, the top-degree homogeneous component $\mathfrak G_w^{\mathrm{top}}$ of the Grothendieck polynomial $\mathfrak G_w$ has attracted increasing attention~\cite{cy24, hafner22, hmss24, py24, psw24}. We emphasize two reasons for its importance. First, the degree of $\mathfrak G_w^{\mathrm{top}}$ determines the \emph{Castelnuovo--Mumford regularity} of matrix Schubert varieties~\cite{rrrsw21}. Second, the support of $\mathfrak G_w$, which is the set of monomials appearing in $\mathfrak G_w$, is conjectured to be governed by the support of $\mathfrak G_w^{\mathrm{top}}$. The main goal of this extended abstract is to make partial progress toward this conjecture.

\begin{conj}[{\cite{mss25}}]
\label{Conj: main}
For a permutation $w$,
\begin{equation}
\label{eqn:master-support}
\supp(\mathfrak G_w)
= \bigcup_{\substack{\alpha \in \supp(\mathfrak S_w) \\ \beta \in \supp(\mathfrak G_w^{\mathrm{top}})}}
[\alpha, \beta],
\end{equation}
where $[\alpha,\beta]$ denotes the component-wise interval
$\{\gamma \in \ZZ^n : \alpha_i \le \gamma_i \le \beta_i \textup{ for all } i\}$.
\end{conj}

Conjecture~\ref{Conj: main} was previously known only for vexillary permutations \cite{hafner22}.
Motivated by this conjecture, we study the polynomial $\mathfrak G_w^{\mathrm{top}}$
and its support. We focus on two families of permutations:

\begin{defn}
A permutation $w$ is called \definition{fireworks} if the initial terms of the decreasing runs of $w$ are increasing. A permutation $w$ is \definition{inverse fireworks} if $w^{-1}$ is fireworks.
\end{defn}
\begin{exa}
    The permutation $w=\underline{3}1\underline{6}\underline{7}542$ is fireworks and the permutation $u=\underline{3}1\underline{7}5\underline{6}42$ is not fireworks. The initial terms of each decreasing run are underlined.
\end{exa}

When $w$ lies in one of these two families, $\mathfrak G_w^{\mathrm{top}}$ exhibits remarkable structure via the 
following two theorems. 

\begin{thm}[\cite{mss25}]
\label{T: MSS}
When $w$ is fireworks, $\mathfrak G_w^{\mathrm{top}}$ consists of a single monomial.  
Specifically, it is a positive integer multiple of $x^{\wt(\overline{D(w)})}$, where $\overline{D(w)}$ denotes the up-closure of the Rothe diagram of $w$ (see Definition~\ref{D: up-closure}).    
\end{thm}

\begin{thm}[\cite{psw24}]
\label{T: PSW}
For all $w\in S_n$, there is a unique inverse fireworks permutation $\pi(w)$ with $\mathfrak G_w^{\mathrm{top}} = c_w \cdot \mathfrak G_{\pi(w)}^{\mathrm{top}}$ for some $c_w\in\ZZ$. In fact, $c_w=|\mathfrak G_{\pi(w^{-1})}^{\mathrm{top}}(1,\dots,1)|$.
\end{thm}  

The map $\pi(\cdot)$ can be computed 
explicitly (see Section~4.2 of~\cite{psw24}).
Consequently, to understand all 
$\mathfrak G_w^{\mathrm{top}}$, 
it suffices to study the case of inverse fireworks permutations. 

Based on these two theorems, we prove several results regarding $\mathfrak G_w^{\mathrm{top}}$ and $\supp(\mathfrak G_w)$ when $w$ is fireworks or inverse fireworks. The following summarized results are based on joint work of the first and second authors on fireworks~\cite{cs25}, and of the first and third authors on inverse fireworks~\cite{cy25}.

\begin{thm}[\cite{cs25}]
\label{thm:fire-main}
Conjecture~\ref{Conj: main} holds when $w$ is fireworks:
\[
\supp(\mathfrak G_w) = \bigcup_{\alpha\in\supp(\mathfrak S_w)}[\alpha,\wt(\overline{D(w)})].
\]
\end{thm}
A surprising corollary of Theorem~\ref{thm:fire-main} is that \emph{homogenized Grothendieck polynomials} $\widetilde{\mathfrak G}_w$ of fireworks permutations have M-convex support. In general, M-convexity of $\supp(\widetilde{\mathfrak G}_w)$ is far from a formal consequence of Conjecture~\ref{Conj: main}; it is even an open problem to show that $\supp(\mathfrak G_w^{\mathrm{top}})$ is M-convex. Conversely, in the case that $\mathfrak S_w$ has zero-one coefficients, $\supp(\widetilde{\mathfrak G}_w)$ is known to be M-convex (\cite{ccmm22}) while Conjecture~\ref{Conj: main} remains open.

\begin{thm}[\cite{cs25}]
\label{thm:m-convex}
For fireworks $w\in S_n$, the homogenized Grothendieck polynomial $\widetilde{\mathfrak G}_w$ has M-convex support. In particular, $\mathfrak G_w$ has SNP, each degree component $\mathfrak G_w^{(c)}$ has M-convex support, and the sizes $|\supp(\mathfrak G_w^{(c)})|$ of their supports form a log-concave sequence.
\end{thm}

Because taking convex hull commutes with homogenization, Theorem~\ref{thm:m-convex} implies that ordinary Grothendieck polynomials $\mathfrak G_w$ of fireworks permutations have SNP.

As far as the authors are aware, M-convexity of $\supp(\widetilde{\mathfrak G}_w)$ was previously known for only exponentially many permutations in $n$ (cf. \cite{fms21}). Theorem~\ref{thm:m-convex} supplies superexponentially many permutations for which $\supp(\widetilde{\mathfrak G}_w)$ is M-convex.

A special subclass of fireworks permutations called \emph{layered} permutations has previously appeared in various maximization problems in Schubert calculus \cite{ms16,mpp19,mppy24,gl24,zhang25}. %\note{A longstanding conjecture of Merzon--Smirnov \cite{ms16} states that the maximal value of the principal specialization $\mathfrak S_w(1,\dots,1)$ is achieved at a layered permutation (see \cite{mpp19}). The maximal value of the principal specialization $\mathfrak G^{(\beta=1)}_w(1,\dots,1)$ of the $\beta=1$ Grothendieck polynomial is asymptotically achieved at a layered permutation \cite{mppy24}. The maximal value of $\deg(\mathfrak G_w) - \deg(\mathfrak S_w)$ is achieved at a layered permutation \cite{psw24}. The size $|\supp(\mathfrak S_w)|$ of the support of $\mathfrak S_w$ is conjecturally maximized at layered permutations \cite{gl24}.
%
%We extend this list with a result of similar flavor.}
The support formula in Theorem~\ref{thm:fire-main} implies that layered permutations maximize Newton polytopes of fireworks Grothendieck polynomials with respect to inclusion.
\begin{cor}[\cite{cs25}]
\label{cor:maximal-newton}
Let $w$ be a fireworks permutation. Then $\pi(w)$ is the layered permutation whose block sizes are the lengths of descending runs of $w$, and $\supp(\mathfrak G_{\pi(w)})\supseteq\supp(\mathfrak G_w)$.
\end{cor}
When $w$ is inverse fireworks, we can prove a weaker version of Conjecture~\ref{Conj: main}, which is~\cite[Conj.~1.2]{mss25}:

\begin{thm}[\cite{cy25}]
\label{thm:invfire-support}
Let $w$ be an inverse fireworks permutation.
If $\mathbf x^\alpha$ appears in $\mathfrak G_w$ with $\deg(\mathbf x^\alpha) < \deg(\mathfrak G_w)$, then there exists $i$ such that $x_i \cdot \mathbf x^\alpha$ also appears in $\mathfrak G_w$.
\end{thm}  

Finally, we define a novel combinatorial object which we call \emph{bumpless vertical-less pipe dreams} (BVPDs), labeled by inverse fireworks permutations. By Theorem~\ref{T: PSW}, BVPDs yield a combinatorial formula for all $\mathfrak G_w^{\mathrm{top}}$.

\begin{thm}[\cite{cy25}]
\label{thm:invfire-main}
For any $w \in S_n$,
\[
\mathfrak G_w^{\mathrm{top}}
= |\BVPD(\pi(w^{-1}))| \cdot \sum_{P \in \BVPD(\pi(w))} \mathbf x^{\wt(P)}.
\]
\end{thm}

As far as the authors are aware, no direct combinatorial formula for $\mathfrak G_w^{\mathrm{top}}$ was previously known. Theorem~\ref{thm:invfire-main} provides the first such formula.

\section{Background}
\label{S: Background}
Our main combinatorial tool is pipe dreams. A {\color{darkred}\emph{pipe dream}} is a filling of a staircase grid $\{(i,j)\in[n]\times[n]\colon i + j \leq n\}$ using {\color{darkred}\emph{cross tiles}} $\ptile$ and {\color{darkred}\emph{bump tiles}} $\bumptile$. By placing half-bump tiles $\jtile$ on the antidiagonal $\{(i,j)\in[n]\times[n]\colon i+j = n+1\}$, a pipe dream can be viewed as a network of $n$ pipes with an associated permutation (see Figure~\ref{fig:pd-mvpd}). Given a pipe dream $P$, there are two different ways to trace pipes in $P$. The {\color{darkred}\emph{top-to-left trace}} is obtained by tracing the pipes from top border to the left, ignoring any repeat crossings between pairs of pipes that has already crossed. The pipe dream's associated permutation is then obtained by reading the pipe indices on the left from top to bottom. The {\color{darkred}\emph{left-to-top trace}} is obtained by tracing the pipes from the left border to the top, again ignoring any repeat crossings between pairs of pipes that have already crossed. The pipe dream's associated permutation is obtained by taking the inverse of the pipe indices on the top read from left to right. It is a well-known fact that, although the paths traced by the two traces may differ, they yield the same permutation.


Let $\PD(w)$ be the set of pipe dreams associated with $w$ and let $\wty(P)$ be the set of its cross tiles $\ptile$. Following~\cite{fk94, km04}, we define the {\color{darkred}\emph{Grothendieck polynomial}} $\mathfrak G_w(\mathbf x)$ to be
\begin{equation}
\label{EQ: Gro formulas}
\mathfrak G_w(\mathbf x) \colonequals \sum_{P \in \PD(w)}  \left((-1)^{|\wty(P)| - \ell(w)}\prod_{(i,j) \in \wty(P)} x_i\right).
\end{equation}
We define the {\color{darkred}\emph{homogenized Grothendieck polynomial}} $\widetilde{\mathfrak G}_w (\mathbf x, z)$ to be
\begin{equation}
\label{EQ: homog Gro formulas}
\widetilde{\mathfrak G}_w(\mathbf x, z) \colonequals \sum_{P \in \PD(w)} \left(z^{\deg(\mathfrak G_w) - |\wty(P)|} \cdot (-1)^{|\wty(P)| - \ell(w)}\prod_{(i,j) \in \wty(P)} x_i\right),
\end{equation}
so that $\widetilde{\mathfrak G}_w$ is homogeneous of degree $\deg(\mathfrak G_w)$.

A \emph{diagram} is a subset of $[n] \times [n]$, 
which can be represented by placing a cell in 
row $i$ column $j$ of an $n \times n$ grid. 
We adopt the convention where
row $1$ is the highest and column $1$ is the leftmost. 
For $w \in S_n$,
its Rothe diagram $D(w)$ 
is the following diagram
$$
D(w) := \{(i, w(j)): 1 \leq i < j \leq n, w(i) > w(j)\}.
$$
\begin{defn}
\label{D: up-closure}
The upward closure of a diagram $D$,
denoted as $\overline{D}$,
is $\{(i,j) \in [n] \times [n]: (i', j) \in D \text{ for some } i' \geq i \}.$
\end{defn}

%For any $w \in S_n$, we define a \emph{marked vertical-less pipe dream} (MVPD) by removing certain pipes from a pipe dream of $w$ while preserving all combinatorial information (see Figure~\ref{fig:pd-mvpd}). 

%MVPDs exhibit two special advantages when $w$ is inverse fireworks, leading to two main results.
%First, we can use droop moves to give a constructive proof of a weaker form of Conjecture~\ref{Conj: main}, which is~\cite[Conj.~1.2]{mss25}:

%Second, when $w$ is inverse fireworks, the MVPDs that contribute to $\mathfrak G_w^{\mathrm{top}}$ admit a simple characterization (see Lemma~\ref{L: Max MVPD}). Let $\widehat{\MVPD}(w)$ be the set of these MVPDs. Using this characterization, we give a bijection from $\widehat{\MVPD}(w)$ to another combinatorial object which we call \emph{bumpless vertical-less pipe dreams} (BVPDs) (see Definition~\ref{D:?}). By Theorem~\ref{T: PSW}, BVPDs yield a combinatorial formula for all top-degree homogeneous component of Grothendieck polynomials.

\section{Fireworks Grothendieck polynomials}
\subsection*{Conjecture~\ref{Conj: main} for fireworks permutations}
For any permutation $w$, 
M\'esz\'aros, Setiabrata and St.\ Dizier~\cite{mss22} showed that every monomial $\mathbf x^\alpha$ appearing in $\mathfrak G_w$ divides $\mathbf x^{\overline{D(w)}}$.
We start by interpreting this result via pipe dreams. Consider the top-to-left trace of a pipe dream $P$ and write $P_{\TL}^{(j)}$ for the pipe $P$ entering in the $j$-th column. The \emph{pipe-weight} of a pipe dream $P \in \PD(w)$ of $w\in S_n$ is the set
\begin{equation}
\label{eqn:pipe-weight}
\pwt(P)\colonequals \{(i,j) \colon \textup{pipe $P_{\TL}^{(j)}$ exits from the bottom edge of a $\ptile$ in row $i$}\}\subseteq[n]^2.
\end{equation}

The pipe weight $\pwt(P)$ of any pipe dream $P\in\PD(w)$ is always a subset of $\overline{D(w)}$. It follows that $\mathbf x^{\wt(P)}$ divides $\mathbf x^{\wt(\overline{D(w)})}$. In the fireworks case, when $\mathbf x^{\wt(\overline{D(w)})}$ appears in $\mathfrak G_w^{\mathrm{top}}$, top-degree pipe dreams have \emph{inclusion-maximal} pipe-weights (rather than just \emph{componentwise-maximal} weights).

The key ingredient in our proof of Theorem~\ref{thm:fire-main} is an explicit pipe dream construction guided by pipe-weights. Specifically, Theorem~\ref{thm:fire-main} follows from repeated application of the following claim.

\begin{thm}[\cite{cs25}]
\label{thm:main2-steps}
Let $w\in S_n$ be fireworks and suppose $P \in \PD(w)$ satisfies $\wt(P) < \wt(\overline{D(w)})$. For any $a$ with $\wt(P)_a < \wt(\overline{D(w)})_a$, there exists a pipe dream $Q \in \PD(w)$ so that $\wt(Q)_b = \wt(P)_b$ for $b<a$, $\wt(Q)_a = \wt(P)_a + 1$, and $\wt(Q)_b \leq \wt(P)_b$ for $b>a$.
\end{thm}

Our proof is local in nature and involves finding configurations like the one in Figure~\ref{fig:main2-steps}, which are guaranteed to exist by pipe-weight considerations (cf.~\eqref{eqn:pipe-weight}).

\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.9]{case-2-fpsac}\vspace{-2ex}
\end{center}
\caption{Left: One type of local configuration in $P\in\PD(w)$ of non-maximal degree. Right: One step of our algorithm constructing desired $Q\in\PD(w)$ for Theorem~\ref{thm:main2-steps}.\vspace{-2ex}}
\label{fig:main2-steps}
\end{figure}

\begin{rem}
Our construction in the proof of Theorem~\ref{thm:main2-steps} yields pipe dreams $Q$ which are equal to $P$ in rows $<a$.
\end{rem}
\newtheorem*{thm:fire-main}{Theorem~\ref{thm:fire-main}}
\begin{thm:fire-main}
Let $w$ be a fireworks permutation. If $\mathbf x^\alpha$ appears in $\mathfrak G_w$ and $x_i\cdot \mathbf x^\alpha$ divides $\mathbf x^{\wt(\overline{D(w)})}$, then $x_i\cdot \mathbf x^\alpha$ appears in $\mathfrak G_w$. In particular,
\[
\supp(\mathfrak G_w) = \bigcup_{\alpha\in\supp(\mathfrak S_w)}[\alpha,\wt(\overline{D(w)})].
\]
\end{thm:fire-main}
\begin{proof}[Proof of Theorem~\ref{thm:fire-main}]
Suppose that $\mathbf x^\alpha$ appears in $\mathfrak G_w$ and that $x_i\cdot \mathbf x^\alpha$ divides $\mathbf x^{\wt(\overline{D(w)})}$. Pick $P\in\PD(w)$ with $\wt(P) = \alpha$. By repeated application of Theorem~\ref{thm:main2-steps}, there exists a pipe dream $Q\in\PD(w)$ with $\mathbf x^{\wt(Q)} = x_i\cdot\mathbf x^\alpha$. The claim follows from Equation~\eqref{EQ: Gro formulas}.
\end{proof}
\subsection*{Schubitopes and M-convexity of $\supp(\mathfrak G_w)$}
We prove Theorem~\ref{thm:m-convex} by decomposing the support of $\mathfrak G_w$ into a Minkowski sum of smaller M-convex sets (Proposition~\ref{prop:psp-integer-points}). We are motivated by the following analogous story for Schubert polynomials.

\begin{defn}
A set $S\subset\ZZ^n$ is {\color{darkred}\emph{M-convex}} if for any $\alpha,\beta\in S$ and any $i\in[n]$ for which $\alpha_i > \beta_i$, there is an index $j\in[n]$ for which $\alpha_j < \beta_j$ and $\alpha - \mathbf e_i + \mathbf e_j \in S$ and $\beta + \mathbf e_i - \mathbf e_j \in S$.
\end{defn}

There is an interpretation of M-convexity in terms of certain polytopes called generalized permutahedra: A set is M-convex if and only if it is the set of integer points of an integral generalized permutahedron. In particular, a polynomial $f$ has M-convex support if and only if $f$ has SNP and its Newton polytope is a generalized permutahedron. We refer to \cite{postnikov09} for background on generalized permutahedra.

\begin{lem}
\label{lem:m-convex-sum}
The Minkowski sum of M-convex sets is M-convex.
\end{lem}

\begin{lem}
\label{lem:log-concavity}
Let $S\subset\ZZ^n$ be an M-convex set, and let $I\subseteq[n]$. Then the sequence
\[
a_c\colonequals \left|S\cap\left\{\sum_{i\in I}x_i = c\right\}\right|
\]
of point counts on the hyperplanes $\left\{\sum_{i\in I}x_i = c\right\}$ is log-concave.
\end{lem}
%\begin{proof}[Proof sketch]
%The generating function $f_S\colonequals \sum_{\alpha\in S}\mathbf x^\alpha$ is \emph{denormalized Lorentzian} (see \cite{bh20,blp23}). By \cite[Lem 4.8]{blp23}, the specialization
%\[
%f_S(\mathbf x)\bigg|_{\substack{x_i \mapsto q \textup{ if } i \in S\\ x_i \mapsto t \textup{ if } i\not\in S}} = \sum_c a_c q^c t^{\deg f_S-c}
%\]
%is again denormalized Lorentzian. Then \cite[Prop 4.4]{bh20} implies that $(a_c)$ is log-concave.
%\end{proof}
\begin{defn}
\label{defn:schubert-matroid-polytope}
Given a subset $S \subseteq[n]$ with elements $s_1 < \dots < s_r$, write $\mathcal B$ for the collection of sets $B\subseteq[s_r]$ with elements $b_1 < \dots < b_r$ satisfying $b_i \leq s_i$ for all $i$. The indicator vectors $\{\zeta_B\colon B\in\mathcal B\}\subset \ZZ^n$ form an M-convex set, and their convex hull $P(\mathrm{SM}(S))\colonequals\conv\{\zeta_B\colon B\in\mathcal B\}$ the {\color{darkred}\emph{Schubert matroid polytope}}.
%
%See \cite[pg 2]{mty19} or \cite[Thm 10]{fms18} for a description of the rank function.
\end{defn}
Schubert matroid polytopes compute the support of Schubert polynomials.

\begin{thm}[{\cite[Thm 4, Thm 7]{fms18}}]
\label{thm:schub-support}
Let $w \in S_n$, and let $D_1, \dots, D_n$ denote the columns of $D(w)$. Then $\mathfrak S_w$ has M-convex support and
\[
\supp(\mathfrak S_w) = \left(\sum_{i=1}^n P(\SM(D_i))\right)\cap\ZZ^n.
\]
\end{thm}

\begin{defn}%[cf.\ Definition~\ref{defn:schubert-matroid-polytope}]
\label{defn:schubert-spanning-set-polytope}
Given a subset $S\subseteq[n]$ with elements $s_1 < \dots < s_r$, write $\mathcal B_{\mathrm{sp}}$ for the {\color{darkred}\emph{spanning sets}} of the Schubert matroid; it is the collection of sets $B'\subseteq[s_r]$ with elements $b_1 < \dots < b_{r'}$, with $r' \geq r$, satisfying $b_i \leq s_i$ for all $i\in[r]$. The convex hull $P_{\mathrm{sp}}(\mathrm{SM}(S))\colonequals \mathrm{conv}(\{\zeta_{B'}\colon B'\in\mathcal B_{\mathrm{sp}}\})$ of the indicator vectors is called the {\color{darkred}\emph{Schubert spanning set polytope}}. The homogenized indicator vectors $\{\widetilde\zeta_{B'}\colon B'\in\mathcal B_{\mathrm{sp}}\}$ form an M-convex set, and their convex hull $\widetilde P_{\mathrm{sp}}(\mathrm{SM}(S))\colonequals\mathrm{conv}(\{\widetilde\zeta_{B'}\colon B'\in\mathcal B_{\mathrm{sp}}\})$ is the {\color{darkred}\emph{homogenized Schubert spanning set polytope}}.


\end{defn}

\begin{prop}[{cf.\ \cite[Thm 3.6]{mss25}}]
\label{prop:psp-integer-points}
Fix subsets $D_1, \dots, D_m \subseteq[n]$ and set $P\colonequals P(\SM(D_1)) + \dots + P(\SM(D_m))$. Then
\[
\left(\sum_{j=1}^nP_{\mathrm{sp}}(\SM(D_j))\right) \cap \ZZ^n = \bigcup_{\beta\in \supp(P)}[\beta,\wt(\overline D)].
\]
\end{prop}

\newtheorem*{thm:m-convex}{Theorem~\ref{thm:m-convex}}
\begin{thm:m-convex}
For fireworks $w\in S_n$, the homogenized Grothendieck polynomial $\widetilde{\mathfrak G}_w$ has M-convex support, i.e., $\widetilde{\mathfrak G}_w$ has SNP and its Newton polytope is a generalized permutahedron. In particular, each degree component $\mathfrak G_w^{(c)}$ has M-convex support, and the sizes $|\supp(\mathfrak G_w^{(c)})|$ of their supports form a log-concave sequence.
\end{thm:m-convex}
\begin{proof}[Proof of Theorem~\ref{thm:m-convex}]
Let $D_j$ denote the $j$-th column of $D(w)$ and write 
\[
\mathcal G_{D(w)}\colonequals \sum_{j=1}^n P_{\mathrm{sp}}(\SM(D_j)), \qquad \widetilde{\mathcal G}_{D(w)}\colonequals \sum_{j=1}^n\widetilde P_{\mathrm{sp}}(\SM(D_j)).
\]
By Theorem~\ref{thm:fire-main}, Theorem~\ref{thm:schub-support}, and Proposition~\ref{prop:psp-integer-points}, we have $\supp(\mathfrak G_w) = \mathcal G_{D(w)} \cap \ZZ^n$.
Homogenizing yields $\supp(\widetilde{\mathfrak G}_w) = \widetilde{\mathcal G}_{D(w)} \cap \ZZ^{n+1}$, and Lemma~\ref{lem:m-convex-sum} implies that $\widetilde{\mathfrak G}_w$ has M-convex support. The final claim follows from Lemma~\ref{lem:log-concavity}.
\end{proof}
\subsection*{Inclusion-maximal support}
We prove Corollary~\ref{cor:maximal-newton} using the support formula of Theorem~\ref{thm:fire-main} to reduce the problem to the following simple claim about inclusions among Schubert spanning set polytopes.

\begin{lem}
\label{lem:psp-inclusion}
Let $A\subseteq B\subseteq[n]$ be two sets satisfying $\max(A) = \max(B)$. Then $P_{\mathrm{sp}}(\SM(B)) \subseteq P_{\mathrm{sp}}(\SM(A))$.
\end{lem}

\begin{defn}
For integers $b_1, \dots, b_m$, the {\color{darkred}\emph{layered permutation}} with \emph{block sizes} $(b_1, \dots, b_m)$ is the longest element of the Young subgroup $S_{b_1}\times S_{b_2}\times\dots\times S_{b_m}$; i.e., it is the permutation with one-line notation
\[
c_1 \, (c_1 - 1)\,\dots \,1\, c_2\, (c_2 - 1)\,\dots\,(c_1 + 1)\, \dots\, c_m\, (c_m - 1)\,\dots\,(c_{m-1}+1),
\]
where $c_i \colonequals b_1 + \dots + b_i$.
\end{defn}


\begin{lem}
\label{lem:tip-inclusion}
Let $w$ be a fireworks permutation and write $\pi(w)$ for the layered permutation whose block sizes are the lengths of descending runs of $w$. Then $D(w)_{w(i)} \supseteq D(\pi(w))_{(\pi(w))(i)}$.
\end{lem}

\newtheorem*{cor:maximal-newton}{Corollary~\ref{cor:maximal-newton}}

\begin{cor:maximal-newton}
Let $w$ be a fireworks permutation and write $\pi(w)$ for the layered permutation whose block sizes are the lengths of descending runs of $w$. Then $\supp(\mathfrak G_{\pi(w)})\supseteq\supp(\mathfrak G_w)$.
\end{cor:maximal-newton}
\begin{proof}[Proof of Corollary~\ref{cor:maximal-newton}]
Write $D_i\colonequals D(w)_{w(i)}$ and $D_i'\colonequals D(\pi(w))_{(\pi(w))(i)}$. Lemma~\ref{lem:tip-inclusion} implies that $D_i \supseteq D_i'$ and $\max(D_i) = \max(D_i')$, so Lemma~\ref{lem:psp-inclusion} gives
\[
\supp(\mathfrak G_w) = \left(\sum_{i=1}^n P_{\mathrm{sp}}(\SM(D_i))\right)\cap\ZZ^n \subseteq \left(\sum_{i=1}^n P_{\mathrm{sp}}(\SM(D_i'))\right)\cap\ZZ^n = \supp(\mathfrak G_{\pi(w)}).\qedhere
\]
\end{proof}



\section{Inverse Fireworks Grothendieck polynomials}
The main result of this section is 
Theorem~\ref{thm:invfire-support}
and a bumpless vertical-less pipe dream formula (Theorem~\ref{thm:invfire-main}). More precisely, we introduce a new set of combinatorial objects $\MVPD(w)$ 
for each $w \in S_n$ 
and define weight-preserving bijections
\begin{align*}
\Phi\colon \PD(w) &\to \MVPD(w) \qquad\textup{(Proposition~\ref{P: PD->MVPD})},\\
\Psi\colon \BVPD(\pi(w)) &\to \hMVPD(\pi(w)) \qquad\textup{(Theorem~\ref{T: BVPD->hMVPD})},
\end{align*}
where $\hMVPD(w)$ denotes the subset of $\MVPD(w)$ of highest degree and $\pi(w)$ is the inverse fireworks permutation guaranteed by Theorem~\ref{T: PSW}.

% Given a pipe dream $P$, there are two different ways to endow $P$ with the data of \emph{pipes} of $P$. The {\color{darkred}\emph{$i$-th top-to-left pipe}} $P_{\TL}^{(i)}$ of $P$ is obtained by tracing the pipe in the $i$-th column of the top border as it travels to the left, ignoring any repeat crossings with pipes $P_{\TL}^{(j)}$ that it has already crossed. The {\color{darkred}\emph{$i$-th left-to-top pipe}} $P_{\LT}^{(i)}$ of $P$ is obtained by tracing the pipe in the $i$-th row of the left border as it travels to the top, ignoring any repeat crossings with pipes $P_{\LT}^{(j)}$ that it has already crossed.

% It is a well-known fact that, although the paths traced by the top-to-left pipes and the left-to-top pipes may differ, the pipe at the $i$-th row of the left boundary hits the top boundary at the same column.

% By labeling the pipes of a pipe dream $P$ with the numbers $1$ through $n$ along the top edge, and transporting these labels through the top-to-left pipes or through the left-to-top pipes, and recording the labels reading down the rows on the left boundary, one obtains the one-line notation of the permutation associated to the pipe dream $P$. For $w\in S_n$, we write $\PD(w)$ for the set of pipe dreams whose associated permutation is $w$.

% \begin{exa}
% See Figure~\ref{fig:tl-vs-lt} for a pipe dream associated to the permutation $w = 3162754$. The top-to-left pipes $P_\TL$ are displayed in the center, and the left-to-top pipes $P_\LT$ are displayed in the right. The pipes $(P_\TL^{(1)}, \dots, P_\TL^{(7)})$ and $(P_\LT^{(1)}, \dots, P_{\LT}^{(7)})$ are colored (red, orange, yellow, green, blue, indigo, violet).
% \begin{figure}[ht]
% \begin{center}
% \includegraphics{tl-vs-lt-v2}
% \end{center}
% \caption{Left: A pipe dream in $\PD(3162754)$. Center: The top-to-left pipes $P_\TL$. Right: The left-to-top pipes $P_\LT$.}
% \label{fig:tl-vs-lt}
% \end{figure}
% \end{exa}






\subsection*{Marked vertical-less pipe dreams}
\begin{defn}
A \definition{marked vertical-less pipe dream} (MVPD) is a filling of an $n \times n$ grid using the tiles
\[
\btile,\quad \quad \htile, \quad \quad \ptile, \quad \quad \jtile, \quad \quad \rtile, \quad \quad \mtile, \quad \quad \bumptile,
\]
such that any pipe in a marked tile $\mtile$ (call it $T$) has a $\htile$ in a lower row than $T$ (see Figure~\ref{fig:pd-mvpd}). The adjective \emph{vertical-less} reflects the absence of vertical tiles $\vtile$ in a MVPD.
\end{defn}

Similar to left-to-top traces of ordinary pipe dreams, we trace pipes of a MVPD from left to top, ignoring double crossings. The pipe entering in row $i$ is denoted $P^{(i)}$.
\begin{defn}
The \definition{column-to-row code} $\alpha(M)$ of an MVPD $M$ is the sequence of $n$ numbers defined as follows. If there is no pipe exiting at column $c$ of $M$, then $\alpha(M)_c = 0$. Otherwise, if the pipe $P^{(r)}$ exits in column $c$, then $\alpha(M)_c = r$.
\end{defn}

\begin{defn}
Let $w \in S_n$. Let $\alpha(w)$ denote the sequence obtained by writing the one-line notation $(w^{-1}(1), \dots, w^{-1}(n))$ of $w^{-1}$ and then replacing the left-to-right maxima with $0$'s. Then define $\MVPD(w)$ to be the set of MVPDs with column-to-row code $\alpha(w)$. Also define $\hMVPD(w)\subseteq\MVPD(w)$ to be the set of MVPDs with the maximal number of weighty tiles among all $M\in\MVPD(w)$.
\end{defn}

When drawing an MVPD, we omit blank rows and columns on the bottom and right.
\begin{exa}
\label{E: MVPD}
Let $w = 12643857$, so that $w^{-1} = 12547386$ and $\alpha(w) = (0,0,0,4,0,3,0,6)$. Figure~\ref{fig:pd-mvpd} consists of a pipe dream $P \in \PD(w)$ and an MVPD $M \in \MVPD(w)$.
\begin{figure}[ht]
\begin{center}
    \begin{tikzpicture}[x=1.5em,y=1.5em,thick,rounded corners,color = blue, scale=0.9]
    \draw[step=1,gray,ultra thin] (0,0) grid (8,8);
     \draw[color=blue, thick] (0,5.5)--(1.5,5.5)--(1.5,7.5)--(3.5,7.5)--(3.5,8);
     \draw[color=blue, thick] (0,4.5)--(0.5,4.5)--(0.5,6.5)--(4.5,6.5)--(4.5,7.5);
     \draw[color=blue, thick] (4.5,7.5)--(5.5,7.5)--(5.5,8);
     \draw[color=blue, thick] (0,2.5)--(0.5,2.5)--(0.5,3.5)--(2.5,3.5)--(2.5,4.5);
     \draw[color=blue, thick](2.5,4.5)--(3.5,4.5)--(3.5,5.5)--(4.5,5.5)--(4.5,6.5)--(6.5,6.5)--(6.5,7.5);
     \draw[color=blue, thick](6.5,7.5)--(7.5,7.5)--(7.5,8);
     \draw[color=red, thick] (0,7.5)--(0.5, 7.5)--(0.5, 8);
     \draw[color=red, thick] (0,6.5)--(0.5, 6.5)--(0.5,7.5)--(1.5, 7.5)--(1.5, 8);
     \draw[color=red, thick] (0,3.5)--(0.5, 3.5)--(0.5,4.5)--(1.5, 4.5)--(1.5, 5.5)--(2.5, 5.5)--(2.5, 8);
     \draw[color=red, thick] (0,1.5)--(0.5, 1.5)--(0.5,2.5)--(1.5, 2.5)--(1.5, 4.5)--(2.5, 4.5);
     \draw[color=red, thick] (2.5, 4.5)--(2.5, 5.5)--(3.5, 5.5)--(3.5, 7.5)--(4.5, 7.5);
     \draw[color=red, thick] (4.5, 7.5)--(4.5, 8);
     \draw[color=red, thick] (0,0.5)--(0.5, 0.5)--(0.5,1.5)--(1.5, 1.5)--(1.5, 2.5)--(2.5, 2.5)--(2.5, 3.5)--(3.5, 3.5)--(3.5, 4.5)--(4.5, 4.5)--(4.5, 5.5)--(5.5, 5.5)--(5.5, 7.5)--(6.5, 7.5);
      \draw[color=red, thick] (6.5, 7.5)--(6.5, 8);
    \end{tikzpicture}\qquad\qquad
    \raisebox{2.7em}{\begin{tikzpicture}[x=1.5em,y=1.5em,thick,rounded corners,color = blue, scale=0.9]
    \draw[step=1,gray,ultra thin] (0,0) grid (8,6);
     \draw[color=blue, thick] (0,3.5)--(1.5,3.5)--(1.5,5.5)--(3.5,5.5)--(3.5,6);
     \draw[color=blue, thick] (0,2.5)--(0.5,2.5)--(0.5,4.5)--(4.5,4.5)--(4.5,5.5)--(5.5,5.5)--(5.5,6);
     \draw[color=blue, thick] (0,0.5)--(0.5,0.5)--(0.5,1.5)--(2.5,1.5)--(2.5,2.5)--(3.5,2.5)--(3.5,3.5)--(4.5,3.5)--(4.5,4.5)--(6.5,4.5)--(6.5,5.5)--(7.5,5.5)--(7.5,6);
     \node at (2.5,2.5) {$\bullet$};
     \node at (4.5,5.5) {$\bullet$};
     \node at (6.5,5.5) {$\bullet$};
    \end{tikzpicture}}
\end{center}\vspace{-1ex}
\caption{A pipe dream $P\in\PD(w)$ and a marked vertical-less pipe dream $M\in\MVPD(w)$, for $w = 12643857$. The MVPD $M$ is equal to $\Phi(P)$.}
\label{fig:pd-mvpd}
\end{figure}
\end{exa}

For an MVPD $M$, let $\wty(M)$ be the set of tiles of $M$ that are $\htile$, $\ptile$ or $\mtile$. We now present a $\wty(\cdot)$-preserving bijection $\Phi\colon\PD(w)\to\MVPD(w)$. Let $P_\LT^{(p_i)}$ denote the pipe that enters from the left in row $i$.

\begin{prop}%[{\cite[Prop 3.6]{cy25}}]
\label{P: PD->MVPD}
Let $w \in S_n$ and let $p_i$ denote the left-to-right maxima of $w^{-1}$. The assignment $\Phi\colon \PD(w) \to \MVPD(w)$ given by removing the pipes $P_\LT^{(p_i)}$ and replacing any resulting $\vhtile$ tiles with $\mtile$ is a $\wty(\cdot)$-preserving bijection (see Figure~\ref{fig:pd-mvpd}).
\end{prop}

Therefore, MVPD is a formula for $\mathfrak G_w$ just like pipe dream. However, a key feature of MVPDs is the presence of local \emph{droop moves}, inherited from bumpless pipe dreams~\cite{lls21,lls23}. Thus, MVPDs serve as an interpolating model that allows certain phenomena from bumpless pipe dreams to be transported into the classical pipe dream setting.

\subsection*{Bumpless vertical-less pipe dreams}

\begin{defn}
A \definition{bumpless vertical-less pipe dream} (BVPD) is a tiling of an $n\times(n-1)$ grid using the tiles
\[
\btile,\quad \quad \htile, \quad \quad \ptile, \quad \quad \jtile, \quad \quad \rtile.
\]
\end{defn}
As with MVPDs, we trace pipes from left to top, and write $P^{(i)}$ for the pipe entering in the $i$-th row.

\begin{defn}
The \definition{BVPD column-to-row code} $\alpha'(M)$ of a BVPD $B$ is the sequence of $n-1$ numbers defined (similarly to the MVPD setting) as follows. If there is no pipe exiting at column $c$ of $B$, then $\alpha'(B)_c = 0$. Otherwise, if the pipe $P^{(r)}$ exits in column $c$, then $\alpha'(B)_c = r$.
\end{defn}

\begin{defn}
Let $w$ be an \emph{inverse fireworks} permutation. Let $\alpha'(w)$ denote the sequence obtained by removing the first entry from $\alpha(w)$. Let $\BVPD(w)$ denote the set of BVPDs with BVPD column-to-row code $\alpha'(w)$.
\end{defn}
When drawing a BVPD, we omit blank rows and columns on the bottom and right.

\begin{exa}
\label{E: BVPD}
See Figure~\ref{fig:bvpd-pd} for a BVPD for $w = 165234$, so that $w^{-1} = 145632$ and $\alpha'(w) = (0,0,0,3,2)$.
\begin{figure}[ht]
\begin{center}
\begin{tikzpicture}[x=1.5em,y=1.5em,thick,rounded corners,color = blue, scale=0.9]
\draw[step=1,white,ultra thin] (0,-3) grid (5,0);
\draw[step=1,gray,ultra thin] (0,0) grid (5,3);
\draw[color=blue, thick] (0,0.5)--(1.5, 0.5)--(1.5, 1.5)--(3.5, 1.5)--(3.5, 3);
\draw[color=blue, thick] (0,1.5)--(0.5, 1.5)--(0.5, 2.5)--(4.5, 2.5)--(4.5, 3);
\end{tikzpicture}
\qquad\qquad
\begin{tikzpicture}[x=1.5em,y=1.5em,thick,rounded corners,color = blue, scale=0.9]
\draw[step=1,gray,ultra thin] (-1,-3) grid (5,3);
\draw[color=red, thick] (-1,2.5)--(-0.5,2.5)--(-0.5,3);
\draw[color=red, thick] (-1,-0.5)--(-0.5,-0.5)--(-0.5,2.5)--(0.5,2.5);
\draw[color=red, thick] (0.5,2.5)--(0.5,3);
\draw[color=red, thick] (-1,-1.5)--(-0.5,-1.5)--(-0.5,-0.5)--(0.5,-0.5)--(0.5,1.5)--(1.5,1.5);
\draw[color=red, thick] (1.5,1.5)--(1.5,3);
\draw[color=red, thick] (-1,-2.5)--(-0.5,-2.5)--(-0.5,-1.5)--(0.5,-1.5)--(0.5,-0.5)--(1.5,-0.5)--(1.5,0.5)--(2.5,0.5)--(2.5,3);
\draw[color=blue, thick] (-1,0.5)--(1.5, 0.5)--(1.5, 1.5);
\draw[color=blue, thick] (1.5, 1.5)--(3.5, 1.5)--(3.5, 3);
\draw[color=blue, thick] (-1,1.5)--(0.5, 1.5)--(0.5, 2.5);
\draw[color=blue, thick] (0.5, 2.5)--(4.5, 2.5)--(4.5, 3);
\end{tikzpicture}
\end{center}\vspace{-2ex}
\caption{A BPVD $B \in \BVPD(w)$ and a pipe dream $P \in \PD(w)$, for $w = 165234$. The pipe dream $P$ turns out to be of highest degree in $\PD(w)$.\vspace{-3ex}}
\label{fig:bvpd-pd}
\end{figure}
\end{exa}

For a BVPD $B$, let $\wty(B)$ be the set of tiles of $B$ that 
are $\ptile$, $\jtile$ or $\htile$. We now present a $\wty(\cdot)$-preserving bijection $\Psi\colon \BVPD(w) \to \hMVPD(w)$ for all inverse fireworks permutations $w$.

\begin{thm}%[{\cite[Thm 4.4]{cy25}}]
\label{T: BVPD->hMVPD}
Let $w\in S_n$ be inverse fireworks. The assignment $\Psi\colon \BVPD(w) \to \hMVPD(w)$ given by adding a column on the left of $B$ consisting of $\htile$ and $\btile$ and then replacing all $\rtile$ tiles with $\mtile$ tiles is a $\wty(\cdot)$-preserving bijection.
\end{thm}

Figure~\ref{fig:bvpd-pd} gives an example of the bijection $\Phi^{-1}\circ\Psi\colon \BVPD(w) \to \PD(w)$. 

Our proof of Theorem~\ref{T: BVPD->hMVPD} relies on the following structural lemma.

\begin{lem}%[{\cite[Lem 4.7]{cy25}}]
\label{L: Max MVPD}
Let $w$ be an inverse fireworks permutation. Then $\hMVPD(w)$ consists of elements in $\MVPD(w)$ without $\rtile$ and $\bumptile$.
\end{lem}

\begin{rem}
The bijections $\Phi$ and $\Psi$ preserve $\wty(\cdot)$, so the formula in Theorem~\ref{thm:invfire-main} works for \emph{double Grothendieck polynomials} of inverse fireworks permutations as well.
\end{rem}

\subsection*{Droop moves and going-up}
In \cite[Defn 5.4]{cy25}, we define \emph{droop moves} $\droop_{(i,j)}(M)$ and $\droop_{(i,j)}'(M)$ on MVPDs, which we use to construct the pipe dreams required to establish Theorem~\ref{thm:invfire-support}. We give two examples of the effect of $\droop_{(i,j)}$ and $\droop_{(i,j)}'$  in Figure~\ref{fig:droop}.

\begin{figure}[ht]
\begin{center}
\begin{tikzpicture}[x=1.5em,y=1.5em,thick,rounded corners,color = black]
\draw[step=1,gray,ultra thin] (0,0) grid (2,5);
\draw[color=blue, thick] (0,1.5)--(2,1.5);
\draw[color=blue, thick] (0,2.5)--(2,2.5);
\draw[color=blue, thick] (0,3.5)--(2,3.5);
\draw[color=blue, thick] (0,0.5)--(0.5,0.5)--(0.5,4.5)--(2,4.5);
\draw[color=blue, thick] (0,4.5)--(.5, 4.5)--(.5, 5);
\draw[color=blue, thick] (1.5, 0)--(1.5, 0.5)--(2,.5);
\node[color=blue] at (1.5,0.5) {$\bullet$};
\node at (-0.5,4.5) {$i$};
\node at (-0.5,0.5) {$i'$};
\node at (0.5,5.5) {$j$};
%\node at (-0.5,0.5) {$a_1$};
%\node at (-0.5,1.5) {$a_2$};
%\node at (-0.5,2.5) {$a_3$};
%\node at (-0.5,3.5) {$a_4$};
%\node at (2.5,1.5) {$b_1$};
%\node at (2.5,2.5) {$b_2$};
%\node at (2.5,3.5) {$b_3$};
%\node at (2.5,4.5) {$b_4$};
\end{tikzpicture}
\quad\raisebox{1.45cm}{$\xrightarrow{\droop_{(i,j)}}$}\quad \hspace{-1em}
\begin{tikzpicture}[x=1.5em,y=1.5em,thick,rounded corners,color = black]
\draw[step=1,gray,ultra thin] (0,0) grid (2,5);
\draw[color=blue, thick] (0,1.5)--(2,1.5);
\draw[color=blue, thick] (0,2.5)--(2,2.5);
\draw[color=blue, thick] (0,3.5)--(2,3.5);
\draw[color=blue, thick] (0,0.5)--(1.5,0.5)--(1.5,4.5)--(2,4.5);
\draw[color=blue, thick] (0,4.5)--(.5, 4.5)--(.5, 5);
\draw[color=blue, thick] (1.5, 0)--(1.5, 0.5)--(2,.5);
\node at (-0.5,4.5) {$i$};
\node at (-0.5,0.5) {$i'$};
\node at (0.5,5.5) {$j$};
\end{tikzpicture}
\qquad\qquad
\begin{tikzpicture}[x=1.5em,y=1.5em,thick,rounded corners,color = black]
\draw[step=1,gray,ultra thin] (0,0) grid (2,5);
\draw[color=blue, thick] (0,1.5)--(2,1.5);
\draw[color=blue, thick] (0,2.5)--(2,2.5);
\draw[color=blue, thick] (0,3.5)--(2,3.5);
\draw[color=blue, thick] (0,0.5)--(0.5,0.5)--(0.5,4.5)--(2,4.5);
\node at (-0.5,4.5) {$i$};
\node at (-0.5,0.5) {$i'$};
\node at (0.5,5.5) {$j$};
%\node at (-0.5,0.5) {$a_1$};
%\node at (-0.5,1.5) {$a_2$};
%\node at (-0.5,2.5) {$a_3$};
%\node at (-0.5,3.5) {$a_4$};
%\node at (2.5,1.5) {$b_1$};
%\node at (2.5,2.5) {$b_2$};
%\node at (2.5,3.5) {$b_3$};
%\node at (2.5,4.5) {$b_4$};
\end{tikzpicture}
\quad\raisebox{1.45cm}{$\xrightarrow{\droop_{(i,j)}'}$}\quad \hspace{-1em}
\begin{tikzpicture}[x=1.5em,y=1.5em,thick,rounded corners,color = black]
\draw[step=1,gray,ultra thin] (0,0) grid (2,5);
\draw[color=blue, thick] (0,1.5)--(2,1.5);
\draw[color=blue, thick] (0,2.5)--(2,2.5);
\draw[color=blue, thick] (0,3.5)--(2,3.5);
\draw[color=blue, thick] (0,0.5)--(1.5,0.5)--(1.5,4.5)--(2,4.5);
\node[color=blue] at (1.5,4.5) {$\bullet$};
\node at (-0.5,4.5) {$i$};
\node at (-0.5,0.5) {$i'$};
\node at (0.5,5.5) {$j$};
\end{tikzpicture}
\end{center}\vspace{-2ex}
\caption{Examples of droop moves $\droop_{(i,j)}$ and $\droop_{(i,j)}'$.}
\label{fig:droop}
\end{figure}

%\begin{defn}
%We say $M\in\MVPD(w)$ is \definition{saturated} if the pipe in any $\rtile$ tile in $M$ is not previously in any $\htile$ tile, and if the two pipes in any $\bumptile$ do not cross.
%\end{defn}


\newtheorem*{thm:invfire-support}{Theorem~\ref{thm:invfire-support}}
\begin{thm:invfire-support}
Let $w$ be an inverse fireworks permutation. If $\mathbf x^\alpha$ appears in $\mathfrak G_w$ and $\deg(\mathbf x^\alpha) < \deg(\mathfrak G_w)$, then there exists $i$ so that $x_i\cdot \mathbf x^\alpha$ appears in $\mathfrak G_w$.
\end{thm:invfire-support}
\begin{proof}[Proof sketch]
Pick $M\in \MVPD(w)$ of weight $\alpha$; necessarily, $M\not\in\hMVPD(w)$. If we can turn a $\bumptile$ into a $\ptile$ or a $\rtile$ into a $\mtile$ and still remain in $\MVPD(w)$, then we are done. Otherwise, we show that there exists $\bumphtile$ or $\rhtile$; in this situation we find and perform a droop move and repeat the whole process.
\end{proof}


\acknowledgements{We thank David Anderson, Ian Cavey, Zachary Hamaker, Tomoo Matsumura, Karola M\'esz\'aros, Alejandro Morales, Greta Panova, Brendon Rhoades, Avery St.~Dizier, and Alexander Yong
for many helpful and inspiring conversations, for careful readings of earlier versions of our work, and for pointing out useful references.}

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