%% if you are submitting an initial manuscript then you should have submission as an option here
%% if you are submitting a revised manuscript then you should have revision as an option here
%% otherwise options taken by the article class will be accepted
\documentclass[finalversion]{FPSAC2025}
\articlenumber{87}

\usepackage{amsmath}%%%
\usepackage{adjustbox}%%%
\usepackage{longtable}%%%


\renewcommand{\emptyset}{\varnothing}
\newcommand{\NN}{\mathbb N}
\newcommand{\QQ}{\mathbb Q}
\newcommand{\RR}{\mathbb R}
\newcommand{\CC}{\mathbb C}
\newcommand{\PP}{\mathbb P}
\newcommand{\ZZ}{\mathbb Z}
\newcommand{\GG}{\mathbb G}
\newcommand{\M}{\mathrm{M}}
\newcommand{\EM}{\widetilde{\mathrm{M}}}
\newcommand{\D}{\mathrm{D}}
\newcommand{\T}{\mathrm{T}}
\newcommand{\kk}{\mathbbm{k}}
\newcommand{\be}{\mathbf e}
\newcommand{\Pos}{\operatorname{Pos}}
\newcommand{\rank}{\operatorname{rank}}
\newcommand{\id}{\mathrm{id}}
\newcommand{\codim}{\operatorname{codim}}
\newcommand{\ind}[1]{\mathbf 1(#1)}
\newcommand{\indnoarg}{\mathbf 1}
\newcommand{\indgp}{\overline{\mathbb I}}
\newcommand{\GP}{\mathsf{GP}}
\newcommand{\ads}{\mathsf{AdS}}
\newcommand{\Phalf}{P'}
\newcommand{\W}{\mathfrak S^B_n}

\newcommand{\face}[2]{\operatorname{face}_{#1}#2}
\newcommand{\lp}{Z}
\newcommand{\stellfan}[1]{\Sigma_{\mathrm{St}_{#1}}}
\newcommand{\debar}{\operatorname{abs}}

\newcommand{\phimap}{\phi^B}
\newcommand{\zetamap}{\zeta^B}

\newcommand{\newword}[1]{\textbf{#1}}

\newcommand{\Oo}{\mathcal{O}(1_o)}
\newcommand{\Oinfty}{\mathcal{O}(1_\infty)}


\newcommand{\hunter}[1]{\textcolor{blue}{HS: #1}}

\theoremstyle{definition}
\newtheorem{thm}{Theorem}[section]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{clm}[thm]{Claim}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{defn}[thm]{Definition}
\newtheorem{propdefn}[thm]{Proposition-Definition}
\newtheorem{conj}[thm]{Conjecture}
\newtheorem{eg}[thm]{Example}
\newtheorem{rem}[thm]{Remark}
\newtheorem{obs}[thm]{Observation}
\newtheorem{exer}[thm]{Exercise}
\newtheorem{question}[thm]{Question}
\newtheorem{maintheorem}{Theorem}	\newtheorem{fact}[thm]{Fact}
\renewcommand{\themaintheorem}{\Alph{maintheorem}}

\numberwithin{equation}{section}

%Schubert
\newcommand{\Sc}{\mathfrak{S}}
%Grothendieck
\newcommand{\Gr}{\mathfrak{G}}
%Arrow with transposition
\newcommand{\ar}[1]
{{\xrightarrow{#1}}}
%%%% notation for divided symmetrization stuff%%%%
\newcommand{\can}{\operatorname{can}}
\newcommand{\trim}{\operatorname{trim}} % trimming/snipping leaves
\newcommand{\blossom}{\operatorname{blossom}} % grow leaves (B operators)
\newcommand{\indexedforests}[1][]{ % set of indexed forests
{\ifx&#1&%
    \operatorname{\mathsf{For}}
\else
    \operatorname{\mathsf{For}}^{#1}
\fi}
}
\newcommand{\forestpoly}[2][]{{\mathfrak{P}_{#2}^{\underline{#1}}}} %m-forest polynomials

\newcommand{\qsym}[2][]{
{\ifx&#1&%
  {\operatorname{QSym}_{#2}}
\else
  {{}^{#1}\!\operatorname{QSym}_{#2}}
\fi}
} % ring of quasisymmetric polynomials

\newcommand{\qseq}[2][]{
{\ifx&#1&%
  {\operatorname{QSeq}_{#2}}
\else
  {{}^{#1}\!\operatorname{QSeq}_{#2}}
\fi}
}

\newcommand{\qsymide}[2][]{
{\ifx&#1&%
  {\operatorname{QSym}_{#2}^+}
\else
  {{}^{#1}\!\operatorname{QSym}_{#2}^+}
\fi}
} % ideal of positive degree quasis
\newcommand{\sym}[1]{\operatorname{Sym}_{#1}} % ring of symmetric polynomials
\newcommand{\symide}[1]{\sym{#1}^+} % ideal of positive degree syms
\newcommand{\lcode}[1]{\operatorname{lcode}(#1)} % lehmer code
\newcommand{\supp}{\operatorname{supp}} % support
\newcommand{\linext}{\operatorname{LinExt}} % linear extensions
\newcommand{\compatible}[2][]{
{\ifx&#1&%
  {\mathcal{C}(#2)}
\else
  {\mathcal{C}^{m}(#2)}
\fi}
} %compatible labelings of indexed forest
\newcommand{\internal}[1]{\operatorname{IN}(#1)} % set of internal nodes
\newcommand{\suchthat}{\;|\;}
\newcommand{\im}[1]{\operatorname{Im}(#1)} % image
\newcommand{\nvect}{\mathsf{Codes}}
\newcommand{\nvectn}[1]{\mathsf{Codes}_{#1}}
\newcommand{\catnvect}{\mathsf{CatCodes}}
\newcommand{\injwords}[1]{\mathrm{Inj}(#1)}

\newcommand{\grass}[1]{\mathsf{Grass_{#1}}} % n-grassmannian

%\author{\texorpdfstring{Philippe Nadeau, Hunter Spink, Vasu Tewari}{Philippe Nadeau, Hunter Spink, Vasu Tewari}}

\newcommand{\vvt}[1]{\textcolor{magenta}{VVT: #1}}
\newcommand{\pn}[1]{\textcolor{red}{PN: #1}}
\newcommand{\hs}[1]{\textcolor{blue}{HS: #1}}

\newcommand{\tcr}[1]{\textcolor{red}{#1}}
\newcommand{\poly}{\operatorname{Pol}} % polynomial ring
\newcommand{\schub}[1]{\mathfrak{S}_{#1}} % schubert polynomials
\newcommand{\red}[1]{\operatorname{Red}(#1)} % reduced words
\newcommand{\des}[1]{\operatorname{Des}(#1)} % descent set
\date{}
\newcommand{\cat}[1]{\operatorname{Cat}_{#1}} % catalan sequence
\newcommand{\lsupp}{\operatorname{lsupp}} %left support

\newcommand{\comp}{\operatorname{comp}} % composition general command
\newcommand{\lr}{\operatorname{lr}} % left right labelings
\newcommand{\hivertpi}{\bm{\pi}} % hivert's isobaric
\newcommand{\hivertpibar}{\overline{\bm{\pi}}} % hivert's isobaric
\newcommand{\idem}{\operatorname{id}} %identity
\newcommand{\slide}[2][]{
{\ifx&#1&%
  {\mathfrak{F}_{#2}}
\else
  {\mathfrak{F}_{#2}^{\underline{#1}}}
\fi}
} % slide polynomial
\newcommand{\particle}[1]{\mathfrak{L}_{#1}} %particle
\newcommand{\chains}{\operatorname{Chains}} % chains in bruhat order
\newcommand{\skewdegpoly}[2]{\mathfrak{D}_{#1,#2}} % skew degree polynomials
\newcommand{\degpoly}[1]{\mathfrak{D}_{#1}} % ordinary degree polynomials
\newcommand{\gz}{\operatorname{GZ}} % Gelfand--Zetlin polytope
\newcommand{\Perm}{\operatorname{Perm}} %permutahedron
\newcommand{\ct}{\operatorname{ev_0}} % constant term
\newcommand{\qdes}[2][]{\operatorname{LTer}_{#1}(#2)} % descent set
\newcommand{\rter}[2][]{\operatorname{RTer}_{#1}(#2)} % rightmost leaves
\newcommand{\tope}[2][]{
{\ifx&#1&%
  {\mathsf{T}_{#2}}
\else
  {\mathsf{T}_{#2}^{\underline{#1}}}
\fi}
} % trimming operator
\newcommand{\bope}[2][]{
{\ifx&#1&%
  {\mathsf{B}_{#2}}
\else
  {\mathsf{B}_{#2}^{(#1)}}
\fi}
} % blossoming operator
\newcommand{\rope}[1]{\mathsf{R}_{#1}} % R-ing operator
\newcommand{\cope}[1]{\mathsf{C}_{#1}} % R-ing operator
\newcommand{\dope}[1]{\mathsf{D}_{#1}} % R-ing operator
\newcommand{\zope}{\mathsf{Z}} % R-ing operator
\newcommand{\nope}{\mathsf{N}} % normalization
\newcommand{\End}{\operatorname{End}} % Endomorphism
\newcommand{\Hom}{\operatorname{Hom}} % Endomorphism
\newcommand{\Trim}[1]{\operatorname{Trim}({#1})} % trimming sequences
\newcommand{\Th}[1][]{\mathsf{ThMon}^{\underline{#1}}} % Thompson
\newcommand{\xR}[1]{\mathsf{xR}_{#1}} %x_iR_i
\newcommand{\Bl}[1]{\mathsf{Bl}({#1})} %Blossoming Representation
\newcommand{\Tr}[1]{\mathsf{Tr}({#1})} %Trimming representation

% m-ary analogues go here

\newcommand{\msig}[2]{{}^{#2}\!\bm{\sigma}_{#1}} % hivert's isobaric
\newcommand{\sfx}{\mathsf{x}} %x but when used as a monomial

%\newcommand{\zigzag}[1]{\zigzag^{#1}} %zigzagforests
\newcommand{\zigzag}[2][]
{
{\ifx&#1&%
  {\mathsf{ZigZag}_{#2}}
\else
  {\mathsf{ZigZag}_{#2}^{#1}}
\fi}
}

\newcommand{\ltfor}[2][] %indexed forests with all leftmost leaves<=n
{
{\ifx&#1&%
  {\mathsf{LTFor}_{#2}}
\else
  {\mathsf{LTFor}_{#2}^{#1}}
\fi}
}

\newcommand{\rtfor}[2][] %indexed forests with all rightmost leaves >n
{
{\ifx&#1&%
  {\mathsf{RTFor}_{>#2}}
\else
  {\mathsf{RTFor}_{>#2}^{#1}}
\fi}
}


\newcommand{\suppfor}[2][] %indexedforestswithsupp<=n
{
{\ifx&#1&%
  {\mathsf{For}_{#2}}
\else
  {\mathsf{For}_{#2}^{#1}}
\fi}
}


\newcommand{\binfor}[1][]{
{\ifx&#1&%
    \operatorname{\mathsf{BinFor}}
\else
    \operatorname{\mathsf{BinFor}}^{#1}
\fi}
} %m binary forests


%\newcommand{\Bin}[1]{\operatorname{Bin}({#1})}
\newcommand{\Bin}[1]{\widehat{#1}}
\newcommand{\hqsym}[2][]{
{\ifx&#1&%
  {\operatorname{HQSym}_{#2}}
\else
  {\operatorname{HQSym}_{#2}^{#1}}
\fi}
} % notation for harmonics of qsym
\newcommand{\fl}[1]{\mathrm{Fl}_{#1}}
\newcommand{\coinv}[1]{\operatorname{Coinv}_{#1}} % coinvariant algebra

%\newcommand{\qscoinv}[1]{\operatorname{QSCoinv}_{#1}} % quasi-coinvariant algebra

\newcommand{\qscoinv}[2][]{
{\ifx&#1&%
  {\operatorname{QSCoinv}_{#2}}
\else
  {{}^{#1}\!\operatorname{QSCoinv}_{#2}}
\fi}
}


%\newcommand{\qscoinv}[1]{\operatorname{QSCoinv}_{#1}} % quasi-coinvariant algebra

\newcommand{\hsym}[1]{\operatorname{HSym}_{#1}} % symmetric harmonics
\newcommand{\cube}{\mathbf{C}} % cubes
\newcommand{\vol}{\operatorname{Vol}} %vol poly
\definecolor{ao}{rgb}{0.0, 0.5, 0.0}
%\newcommand{\col}{\operatorname{col}} % coloring of the forest
\newcommand{\col}{\overline{\rho}_F} % coloring of the forest
\newcommand{\deri}[1]{\operatorname{D}_{#1}} %ordinary derivative
\newcommand{\vope}[2][]{
{\ifx&#1&%
  {\mathsf{V}_{#2}}
\else
  {\mathsf{V}_{#2}^{\underline{#1}}}
\fi}
} % blossoming operator

\newcommand{\paths}[1]{\operatorname{Paths}(#1)}
\newcommand{\ins}{\varepsilon} % this map used to be called iota (can change it back but I have renamed it to align with Bergeron and Sottile)
\newcommand{\degreepoly}{\mathfrak{D}} % used to be mathcal D but altered to align with Postnikov and Stanley
\newcommand{\stlr}{stLR}
\newcommand{\sfc}{\mathsf{c}}
\newcommand{\sfd}{\mathsf{d}}






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%%%\newtheorem{thm}{Theorem}
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\usepackage{lipsum}

%% define your title in the usual way
\title[Quasisymmetric Divided Differences]{Quasisymmetric Divided Differences}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
%% for more than 3 names, use the Oxford comma: One Author, Two Author, Red Author, \and Blue Author
\author{\texorpdfstring{Philippe Nadeau\addressmark{1}, Hunter Spink \thanks{\href{mailto:hunter.spink@utoronto.ca}{hunter.spink@utoronto.ca}. PN was partially supported by French ANR grant ANR-19-CE48-0011 (COMBIN\'E). HS and VT acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), respectively [RGPIN-2024-04181] and [RGPIN-2024-05433]. }\addressmark{2}, \and Vasu Tewari\addressmark{3}}{Philippe Nadeau, Hunter Spink , and Vasu Tewari}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Universite Claude Bernard Lyon 1, CNRS, Ecole Centrale de Lyon, INSA Lyon, Université Jean Monnet, ICJ UMR5208, 69622 Villeurbanne, France\\\addressmark{2,3}Department of Mathematics,
University of Toronto, Toronto, ON M5S 2E4, Canada}

%% put the date of submission here
\received{\today}

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%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{We develop a quasisymmetric analogue of the combinatorial theory of Schubert polynomials. Indexed binary forests play the role of permutations, and the associated divided difference operators compose according  to the ``Thompson monoid'' governing the Thompson group. Our main application of our theory is to study the ring of quasisymmetric coinvariants and the associated quasisymmetric harmonics spaces. In followup work we describe an algebraic-geometric framework which matches the combinatorial theory.}

%% put your French abstract here, or comment this out if you don't have one
\resume{Nous d\'eveloppons un analogue quasisym\'etrique de la th\'eorie combinatoire des polyn\^omes de Schubert. Les for\^ets binaires index\'es jouent le r\^ole des permutations, et les op\'erateurs de diff\'erences divisées associ\'es se composent comme le «~mono\"ide de Thompson~» correspondant  au groupe de Thompson. La principale application de notre th\'eorie est l’\'etude de l’anneau de covariants quasi-sym\'etriques et des espaces harmoniques quasi-sym\'etriques associ\'es. Dans un travail ult\'erieur, nous d\'ecrirons le pendant en g\'eométrie alg\'ebrique de cette th\'eorie combinatoire.}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{Quasisymmetry, Schubert calculus}

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\usepackage[backend=bibtex, giveninits]{biblatex}
\addbibresource{main.bib}
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\begin{document}

\maketitle
%% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above

\section{Introduction}
The ring of quasisymmetric functions $\qsym{}$, first introduced in Stanley's thesis \cite{StThesis} and further developed by Gessel \cite{Ges84}, is ubiquitous throughout combinatorics; see \cite{ABS06} for a high-level explanation and \cite{grinberg2020hopf} for thorough exposition. 
Truncating to finitely many variables $\{x_1,\dots,x_n\}$ gives the ring of \emph{quasisymmetric polynomials} $\qsym{n}\subset \ZZ[x_1,\dots,x_n]$: these are the polynomials such that for any sequence $a_1,\ldots,a_k\ge 1$, the coefficients of $x_{i_1}^{a_1}\cdots x_{i_k}^{a_k}$ and $x_{j_1}^{a_1}\cdots x_{j_k}^{a_k}$ are equal whenever $1\le i_1<\cdots < i_k \le n$ and $1\le j_1<\cdots < j_k \le n$.

We let $\poly_{n}:= \ZZ[x_1,\dots,x_n]$. Letting $\symide{n}$ denote the ideal in $\poly_{n}$ generated by positive degree homogenous symmetric polynomials, the coinvariant algebra $\coinv{n}:= \poly_{n}/\symide{n}$ has been a central object of study for the past several decades. 
An important reason for this is its distinguished basis of Schubert polynomials \cite{LS82} and the divided difference operators \cite{BGG73} that interact nicely with this family-- see \cite{BS98, BJS93, BKTY04, FK96, FS94, HPW22, KM05, LeSo03} for a sampling of the combinatorics underlying this story.
In fact Schubert polynomials lift to a basis of $\poly:= \ZZ[x_1,x_2,\dots]$.
The close relationship between the  combinatorics of symmetric and quasisymmetric polynomials leads to the natural question, first posed in \cite{ABB04}, of what can be said about the analogous quotient $\qscoinv{n}:= \poly_n/\qsymide{n}$, where $\qsymide{n}$ is the ideal generated by positive degree homogenous quasisymmetric polynomials?
\smallskip

In this paper we develop a quasisymmetric analogue of the \textbf{combinatorial theory} of Schubert polynomials $\schub{w}$ and the divided differences $\partial_i=\frac{f-s_i\cdot f}{x_i-x_{i+1}}$ which recursively generate them. 
The reader well-versed with the classical story should refer to \Cref{tab:sym_vs_qsym} for a comparison.
The role of Schubert polynomials $\schub{w}$ is played by the \emph{forest polynomials} $\forestpoly{F}$ of \cite{NT_forest}, and the role of the $\partial_i$ operators are played by certain new \emph{trimming operators} $\tope{i}$.
Just as Schubert polynomials generalize Schur polynomials, the forest polynomials generalize fundamental quasisymmetric polynomials, a distinguished basis of $\qsym{n}$.
The duality between compositions of trimming operators and forest polynomials allows us to expand any polynomial in the basis of forest polynomials.
In fact, a special case of our framework gives a remarkably simple method for directly extracting the coefficients of the expansion of a quasisymmetric polynomial in the basis of fundamental quasisymmetric polynomials.

The interaction between forest polynomials and trimming operators descends nicely to quotients by $\qsymide{n}$, and we thus obtain a basis comprising certain forest polynomials for $\qscoinv{n}$ as well.
Our techniques are robust enough to gain a complete understanding even in the case one quotients by homogenous quasisymmetric polynomials of degree at least $k$ for any $k\geq 1$. \textbf{In the longer form version of this paper} \cite{NST_a}, we use this theory to construct $\qsymide{n}$-harmonics, which turn out to have a basis given by  the volume polynomials of certain polytopes, answering a question of Aval--Bergeron--Li \cite{ABL10}. Therein we also generalize the theory to $m$-ary forests.
\begin{rem}
    In \Cref{sec:table}, see  \Cref{tab:foresttable} for a list of small forest polynomials by forest code $c(F)$ and \Cref{tab:sym_vs_qsym} for a side by side comparison of our theory to the symmetric story.
\end{rem}


In \cite{ NST_3} we investigate the underlying \textbf{geometric theory}, drawing upon the geometric significance of the ordinary divided difference operator.

\section{Quasisymmetric polynomials via \texorpdfstring{$\rope{i}$}{Ri} and \texorpdfstring{$\tope{i}$}{Ti}}
\label{sec:quasisymmetric_polynomials}
%%%%%%%%%%%%%%%%%%%%%%%%%%
The following result is at the heart of our understanding of quasisymmetric polynomials. This characterization does not seem to be widely known, although it was implicitly used in the study of the connection between quasisymmetric functions and James spaces by Pechenik--Satriano \cite{PeSa22}.
We call this the \emph{Bergeron--Sottile map} as they were the first to introduce this \cite{BS98} in the context of  Schubert calculus (see also \cite{BS02, LSS06}).

%%%%%%%%%%%%%
\begin{defn}
    For $f\in \poly_n$ and $i\in\{1,\ldots,n\}$ we define the $i$'th \emph{Bergeron--Sottile map}
    \begin{align}\rope{i}f=f(x_1,\ldots,x_{i-1},0,x_{i},\ldots,x_{n-1}).\end{align}
\end{defn}
%%%%%%%%%%%%%
In other words, $\rope{i}$ sets $x_i=0$ and shifts $x_j\mapsto x_{j-1}$ for all $j\ge i+1$. Note that $\rope{i}$ is well-defined on $\poly$. The following is then an elementary reformulation of quasisymmetry that allows us to work with the condition algebraically.
\begin{thm}
\label{thm:Rqsymchar}
    Let $f\in \poly_n$. Then $f\in \qsym{n}$ if and only if $\rope{1}f=\cdots =\rope{n}f$.
\end{thm}
The typical proof that $\qsym{n}$ is closed under multiplication involves identifying an explicit basis whose multiplication can be explicitly computed. With $\rope{i}$ this is immediate as the equalizer of the algebra maps $\rope{i},\rope{i+1}$ is a subring of $\poly_n$, and the intersection of subrings is a subring.
\begin{cor}
    $\qsym{n}$ is a subring of $\poly_n$.
\end{cor}
%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%
Using these operators we can define our quasisymmetric divided difference operator, which we also dub {\em trimming} operator for reasons that will become clear shortly. 

\begin{defn}
\label{defn:topem}
We define the operator $\tope{i}:\poly\to \poly$ by any of the equivalent expressions
\begin{align}
\tope{i}f:= \rope{i}\partial_if=\rope{i+1}\partial_if=\frac{\rope{i+1}f-\rope{i}f}{x_i}.
\end{align}
\end{defn}

This is the quasisymmetric divided difference at the core of this work. Because $\tope{i}(f)=0$ if and only if $\rope{i+1}(f)=\rope{i}(f)$, we deduce the following.

\begin{thm}
\label{thm:Tqsymchar}
    $f\in \poly_n$ is quasisymmetric if and only if $\tope{1}f=\cdots =\tope{n-1}f=0$.
\end{thm}
%%%%%%%%%%%%%%%%

\section{Forests and the Thompson monoid}
\label{sec:forest_thompson}
A \emph{binary tree} $T$ has internal nodes $v\in \internal{T}$ that have two children $v_L$ and $v_R$, and leaf nodes that have no children. We define $|T|=|\internal{T}|$, and let $\ast$ be the unique tree with $|\ast|=0$. An \emph{indexed forest} $F$ is an infinite sequence  $T_1,T_2,\ldots$ of binary trees where all but finitely many of the trees are $\ast$. We write $\indexedforests$ for the set of all indexed forests.

%Leaves of an indexed forest are naturally labeled by 
There is a monoid structure on $\indexedforests$ where we define $F\cdot G\in\indexedforests$ by identifying the $i$th leaf of $F$ with the $i$th root node of $G$. The empty forest $\emptyset\in\indexedforests$ is the identity element.

\begin{figure}[!ht]
    \centering
    \includegraphics[width=0.8\textwidth]{monoid_multiplication.pdf}
    \caption{The products $F\cdot G$ and $G\cdot F$ for $F,G\in \indexedforests$, with both roots and leaves labeled}
    \label{fig:monoid_multiplication}
\end{figure}

\begin{defn}
    The Thompson monoid is the monoid
    $$\Th:=\langle 1,2,\ldots \suchthat i\cdot j=j\cdot (i+1)\text{ for all }i>j\rangle.$$
\end{defn}
This monoid is both left and right cancellative, and its fraction group is naturally identified with Thompson's group $F$ which has the standard presentation
$$\langle x_1,x_2,\ldots \suchthat x_j^{-1}x_ix_j=x_{i+1}\text{ for }i>j\rangle$$
via the map $i\mapsto x_i^{-1}$. It is a folklore fact in the theory of Thompson groups that
$\Th\cong \indexedforests$
under the map sending $$i\mapsto \underline{i}:= \underbrace{\ast\cdots \ast}_{i-1} \cdot \wedge \cdot \ast \cdot\ast \cdots$$
where $\wedge$ is the unique binary tree with $|\wedge|=1$.

\begin{thm}
The map $i\mapsto \tope{i}$ is a representation of the Thompson monoid on $\poly$. Equivalently,
    $$\tope{i}\tope{j}=\tope{j}\tope{i+1}\text{ for all }i>j.$$
\end{thm}
Because of this, it makes sense to define $\tope{F}$ for $F\in \indexedforests$ by declaring
$$\tope{F}:=\tope{i_1}\cdots \tope{i_k}\text{ for any factorization }F=\underline{i_1}\cdots \underline{i_k}.$$

For a permutation $w$ the descent set is the set of all $i$ such that $w(i)>w(i+1)$. The analogue for forests is the \emph{left terminal set} $\qdes{F}$. We call an internal node $v\in \internal{F}$ \emph{terminal} if both of $v$'s children are leaves, and we say that $i\in \qdes{F}$ if there is a terminal node with left child $i$. This happens if and only if we can write $F=(F/i)\cdot i$ for a (necessarily unique) forest $F/i$, and so
$$\qdes{F}=\{i\suchthat F/i\text{ exists}\}.$$
A reduced word for $w$ is a sequence $(i_1,\ldots,i_{\ell(w)})$ such that $w=s_{i_1}\cdots s_{i_{\ell(w)}}$. Analogously, the set of \emph{trimming sequences} for $F$ is defined as
$$\Trim{F}:=\{(i_1,\ldots,i_{|F|})\suchthat F=\underline{i_1}\cdots \underline{i_{|F|}}\}.$$
Finally, we note that there is a \emph{code map} $c(F)=(a_1,a_2,\ldots)$ where $a_i$ is the number of internal nodes whose leftmost leaf descendent is the $i$'th leaf. Then $i\in \qdes{F}$ is equivalent to $a_i>0$ and $a_{i+1}=0$, and $F\mapsto F/i$ is given by
$(a_1,\ldots,a_{i-1}a_i,0,a_{i+1},\ldots)\mapsto (a_1,\ldots,a_{i-1},a_i-1,a_{i+1},\ldots)$.

We will list small forest polynomials by code in \Cref{sec:table}.

\section{Forest polynomials and Trimming operations}
\label{section:forest_trimming}

Let $S_{\infty}=\bigcup S_n$, the set of permutations of $\NN$ with all but finitely many points fixed. We recall that Schubert polynomials are the unique family of homogenous polynomials indexed by $S_{\infty}$ such that $\schub{id}=1$ and $\partial_i\schub{w}=\delta_{i\in \des{w}}\schub{ws_i}$. They form a basis of $\poly$, those with $\des{w}\subset \{1,\ldots,n\}$ are a $\ZZ$-basis for $\poly_n$; those with $\des{w}\subset \{1,\ldots,n\}$ and $w\not\in S_n$ are a $\ZZ$-basis for $\symide{n}$; and the Schubert polynomials with $w\in S_n$ are a $\ZZ$-basis for $\coinv{n}:=\poly_n/\symide{n}$. Using our theory we arrive at analogous conclusions.

\begin{thm}
\label{thm:forest_polynomials}
    There is a unique family of homogenous polynomials $\left(\forestpoly{F}\right)_F$ indexed by $F\in \indexedforests$ such that $\forestpoly{\emptyset}=1$ and
    $$\tope{i}\forestpoly{F}=\delta_{i\in \qdes{F}}\forestpoly{F/i}.$$
    These are the \textbf{forest polynomials} of the first and third authors \cite{NT_forest}.
   Furthermore,
    \begin{itemize}
        \item The forest polynomials are a $\ZZ$-basis for $\poly$.
        \item The forest polynomials with $\qdes{F}\subset \{1,\ldots,n\}$ are a $\ZZ$-basis for $\poly_n$.
        \item The forest polynomials with $\qdes{F}\subset \{1,\ldots,n\}$ and $\supp{F}\not\subset \{1,\ldots,n\}$ are a $\ZZ$-basis for $\qsymide{n}$.
        \item The forest polynomials with $\supp{F}\subset \{1,\ldots,n\}$ are a $\ZZ$-basis for $\qscoinv{n}=\poly_n/\qsymide{n}$.
    \end{itemize}
\end{thm}
\begin{cor}
\label{cor:forest_decomposition}
    Every $f\in \poly$ can be uniquely written as
    $$f=\sum_{F\in \indexedforests}a_F\forestpoly{F}\,\,\,\text{  where  }\,\,\,a_F=(\tope{F}f)(0,0,\ldots).$$
\end{cor}
\begin{rem}
    The monomial expansion of forest polynomials doesn't play any role in our theory, and no simple seed generates all forest polynomials unlike with Schubert polynomials. For an explicit definition see \cite{NT_forest}, and \cite{NST_2} for further discussion, and see \Cref{sec:table} for a list of forest polynomials.
\end{rem}

\section{Algebraically decomposing \texorpdfstring{$\qsym{n}$}{QSymn} into fundamentals}
\label{sec:fundamentals}
Forests $F$ with $\qdes{F}\subset \{n\}$ play an analogous role to the $n$-Grassmannian permutations (those permutations with $\des{w}\subset \{n\}$) which index the Schur polynomials $s_\lambda$. We call these forests \emph{Zigzag forests}, and denote the set of them $\zigzag{n}$. For an integer sequence $a=(a_1,\ldots,a_k)$ with $a_i\ge 1$ we define the set of \emph{compatible sequences} \begin{align*}\compatible{a}:=\{(i_1,\ldots,i_k):a_j\ge i_j\ge i_{j+1},\text{ and if }a_j>a_{j+1}\text{ then }i_j>i_{j+1}\}.\end{align*}
  Given a sequence $\mathbf{i}=(i_1,\dots,i_k)$ we denote $\sfx_{\mathbf{i}}:=x_{i_1}\cdots x_{i_k}$.
  Let $\qseq{n}$  be the set of sequences $(a_1,\ldots,a_k)$ satisfying $n=a_1\ge \cdots \ge a_k\ge 1$
        and $a_i-a_{i+1}\le 1$ for $1\leq i\leq k-1$.
If $(a_1,\ldots,a_k)\in \qseq{n}$ then
\begin{align*}
    \slide{a}=\sum_{\mathbf{i}\in \compatible[m]{a}}\sfx_{\mathbf{i}}.
\end{align*} is called a \emph{fundamental quasisymmetric polynomial}.
\begin{thm}
    The forest polynomials with $F\in \zigzag{n}$ are a $\ZZ$-basis for $\qsym{n}$, and coincide with the fundamental quasisymmetric polynomials: the mapping $(a_1,\ldots,a_k)\mapsto F=a_k\cdots a_1$ is a bijection
$\qseq{n}\to \zigzag{n}$ under which we have $\slide{a}=\forestpoly{F}$.
\end{thm}
\begin{eg}
\label{eg:fundexpansion}
    Suppose we want to decompose the  quasisymmetric polynomial 
    {\parfillskip=0pt \par} \noindent
    $f(x_1,x_2,x_3)=2x_1^2x_2+2x_1^2x_3+2x_2^2x_3+x_1x_2^2+x_1x_3^2+x_2x_3^2 \in \qsym{3}$ into fundamental quasisymmetrics. Using \Cref{cor:forest_decomposition}, we track in \Cref{fig:qsym_expansion_eg} the nonzero applications $\tope{i_3}\tope{i_2}\tope{i_1}f$ where $(i_1,i_2,i_3)\in \qseq{3}$, and read off $f=\slide{332}+2\slide{322}-3\slide{321}$.
    \begin{figure}[!ht]
        \centering
        \includegraphics{qsym_expansion_eg.pdf}
        \caption{Trimming $f\in \qsym{3}$}
        \label{fig:qsym_expansion_eg}
    \end{figure}
\end{eg}

\section{Diagrammatics}
We define the \emph{quasisymmetric nil-Hecke algebra} to be the subalgebra of $\End(\poly_n)$ generated by multiplication by $\poly$, $\tope{i}$, and $\rope{i}$. Many of the relations between these operations are \emph{formal commutation relations}. Write $\poly_n=V^{\otimes n}$ where $V=\poly_1$. Then we can write
\begin{align*}\text{multiplication by $x_i$}&=\idem^{\otimes i-1}\otimes x\otimes \idem^{\otimes n-i},\\
\rope{i}&=\idem^{\otimes i-1}\otimes \rope{}\otimes \idem^{\otimes n-i},\text{ and }\\
\tope{i}&=\idem^{\otimes i-1}\otimes \tope{}\otimes \idem^{\otimes n-i-1},
\end{align*}
where
$x:\poly_1\to \poly_1$ is multiplication by $x$, $\rope{}:\poly_1\to \poly_0=\ZZ$ is the map $f\mapsto f(0)$, and $\tope{}:\poly_2\to \poly_1$ is the map $f(x,y)\mapsto \frac{f(z,0)-f(0,z)}{z}$. 

We can represent these via diagrams which we should think of taking inputs on the bottom to outputs on the top, and compose two operations $F\circ G$ by stacking the diagram for $F$ on top of the diagram for $G$. Representing $\rope{i}$ and $\tope{i}$ as in \Cref{fig:TRx_diagrams}, and putting a dot on a strand to represent multiplication by $x$, isotopy classes of diagrams give the same operations and encode all \emph{formal commutation relations}. These are the Thompson monoid relations $\tope{i}\tope{j}=\tope{j}\tope{i+1}\text{ for }i>j$ as well as the relations $\rope{i}\rope{j}=\rope{j}\rope{i+1}\text{ for }i\ge j,$ $\tope{i}\rope{j}=\rope{j}\tope{i+1}\text{ for }i\geq j+2$,  
$\tope{i}\rope{j}=\rope{j-1}\tope{i}\text{ for }i\leq j$, $x_ix_j=x_jx_i$, $\tope{i}x_j=x_j\tope{i}$ for $i\ge j+1$, $\tope{i}x_j=x_{j-1}\tope{i}$ for $i \le j-2$, $\rope{i}x_j=x_j\rope{i}$ for $i\ge j+1$, and $\rope{i}x_j=x_{j-1}\rope{i}$ for $i \le j-1$.

\begin{figure}[!ht]
    \centering
    \includegraphics[width=0.9\textwidth]{TR_new}
    \caption{Diagram generators for the quasisymmetric nil-Hecke algebra} 
    \label{fig:TRx_diagrams}
\end{figure}
\begin{thm}
    The nontrivial relations in the quasisymmetric nil-Hecke algebra are $\rope{}x=0$, $\tope{}(x\otimes \idem)=\idem\otimes \rope{}$ and $\tope{}(\idem \otimes x)=-\rope{}\otimes \idem$, and $x\tope{}=\idem\otimes \rope{}-\rope{}\otimes \idem$,
    which have the diagrammatics of \Cref{fig:TRx_diagram_relations}.

    \begin{figure}[!ht]
    \centering    \includegraphics[width=\textwidth]{TRx_relations}
    \caption{Diagram relations for the quasisymmetric nil-Hecke algebra}
    \label{fig:TRx_diagram_relations}
\end{figure}
\end{thm}

\section{Positive multiplication and Schubert decompositions}
Recall that for Schubert polynomials, it is wide open to establish the nonnegativity of the generalized Littlewood--Richardson coefficients arising in the expansion
$$\schub{u}\schub{w}=\sum_{v\in S_n}c^v_{u,w}\schub{v}$$
in a combinatorial manner. The primary difficulty is that the Leibniz rule
$$\partial_i(fg)=\partial_i(f)g+(s_i\cdot f)\partial_i(g)$$
involves the non-Schubert positive operation $f\mapsto s_i\cdot f$. However, for forest polynomials certain straightening rules for these operations allow us to bypass these problems.
\begin{lem}[Twisted Leibniz rule]
\label{lem:mleibniz}
For $f,g\in \poly$ we have
\begin{align*} 
\tope{i}(fg)=\tope{i}(f)\rope{i+1}(g)+\rope{i}(f)\tope{i}(g).
\end{align*}
\end{lem}
\begin{thm}
We have \begin{align*}\rope{i}\forestpoly{F}&=\sum b^G_{i,F}\forestpoly{G}\text{ with }b^G_{i,F}\ge 0;\\
\forestpoly{F}\forestpoly{G}&=\sum c^H_{F,G}\forestpoly{H}\text{ with }c^H_{F,G}\ge 0;\\
\schub{w}&=\sum_{F\in \indexedforests}a^{F}_w\forestpoly{F}\text{ with }a^{F}_w\ge 0.\end{align*}
\end{thm}
\begin{proof}[Proof sketch]
Denote by $\ct f=f(0,0,\ldots)$, the constant term map. Suppose $\Phi$ is a composite of $\rope{i}$ and $\tope{j}$ operations in some order. The relations
$$\tope{i}\rope{j}=\begin{cases} \rope{j}\tope{i+1}&\text{ if }i\ge j+2\\
\rope{i+1}\tope{i}+\rope{i}\tope{i+1}&\text{ if }i=j+1\\
\rope{j-1}\tope{i}&\text{ if }i\le j\end{cases}$$
allow us to iteratively ``move $\rope{j}$ to the left'' in any composite it appears in, and then together with the relation $\ct\rope{j}=\ct$ we conclude
\begin{equation}\label{eqn:Phipositive}\ct\Phi=\sum d^{\Phi}_K\ct\tope{K}\text{ with }d^{\Phi}_K\ge 0.\end{equation}

We see that $b_{i,F}^G=\ct \tope{G}\rope{i}\forestpoly{F}\ge 0$ by applying \eqref{eqn:Phipositive} $\ct\tope{K}\forestpoly{G}=\delta_{K,G}$. To see $c^H_{F,G}=\ct\tope{H}(\forestpoly{F}\forestpoly{G})\ge 0$, we apply induction on $|F|+|G|$ together with the twisted Leibniz rule and \eqref{eqn:Phipositive} on each of the resulting terms. Finally, by induction on $\ell(w)$ we have for any $i\in \qdes{F}$ that  $a^F_w=\ct\tope{F}\schub{w}=\ct\tope{F/i}\tope{i}\schub{w}=\ct\tope{F/i}\rope{i}\partial_i\schub{w}=\delta_{i\in \des{w}}\ct\tope{F/i}\rope{i}\schub{ws_i}$, and we conclude by induction and \eqref{eqn:Phipositive}.
\end{proof}

\acknowledgements{We would like to thank Dave Anderson, Nantel Bergeron, Lucas Gagnon, Darij Grinberg, Allen Knutson, Cristian Lenart, Oliver Pechenik, and Frank Sottile for several stimulating conversations/correspondence.}

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\pagebreak
\section{Tables}
\label{sec:table}
\begin{center}
\footnotesize
\begin{longtable}{|c|l|} 
\hline
$c(F)$ & $\forestpoly{F}$  \\
\hline
$ (0, 0, 0, 0, 0) $ & $ 1 $\\ 
$ (1, 0, 0, 0, 0) $ & $ x_{1} $\\ 
$ (0, 1, 0, 0, 0) $ & $ x_{1} + x_{2} $\\ 
$ (0, 0, 1, 0, 0) $ & $ x_{1} + x_{2} + x_{3} $\\ 
$ (0, 0, 0, 1, 0) $ & $ x_{1} + x_{2} + x_{3} + x_{4} $\\ 
$ (2, 0, 0, 0, 0) $ & $ x_{1}^{2} $\\ 
$ (1, 1, 0, 0, 0) $ & $ x_{1} x_{2} $\\ 
$ (1, 0, 1, 0, 0) $ & $ x_{1}^{2} + x_{1} x_{2} + x_{1} x_{3} $\\ 
$ (1, 0, 0, 1, 0) $ & $ x_{1}^{2} + x_{1} x_{2} + x_{1} x_{3} + x_{1} x_{4} $\\ 
$ (0, 2, 0, 0, 0) $ & $ x_{1}^{2} + x_{1} x_{2} + x_{2}^{2} $\\ 
$ (0, 1, 1, 0, 0) $ & $ x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} $\\ 
$ (0, 1, 0, 1, 0) $ & $ x_{1}^{2} + 2 x_{1} x_{2} + x_{2}^{2} + x_{1} x_{3} + x_{2} x_{3} + x_{1} x_{4} + x_{2} x_{4} $\\ 
$ (0, 0, 2, 0, 0) $ & $ x_{1}^{2} + x_{1} x_{2} + x_{2}^{2} + x_{1} x_{3} + x_{2} x_{3} + x_{3}^{2} $\\ 
$ (0, 0, 1, 1, 0) $ & $ x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} + x_{1} x_{4} + x_{2} x_{4} + x_{3} x_{4} $\\ 
$ (3, 0, 0, 0, 0) $ & $ x_{1}^{3} $\\ 
$ (2, 1, 0, 0, 0) $ & $ x_{1}^{2} x_{2} $\\ 
$ (2, 0, 1, 0, 0) $ & $ x_{1}^{2} x_{2} + x_{1}^{2} x_{3} $\\ 
$ (2, 0, 0, 1, 0) $ & $ x_{1}^{3} + x_{1}^{2} x_{2} + x_{1}^{2} x_{3} + x_{1}^{2} x_{4} $\\ 
$ (1, 2, 0, 0, 0) $ & $ x_{1} x_{2}^{2} $\\ 
$ (1, 1, 1, 0, 0) $ & $ x_{1} x_{2} x_{3} $\\ 
$ (1, 1, 0, 1, 0) $ & $ x_{1}^{2} x_{2} + x_{1} x_{2}^{2} + x_{1} x_{2} x_{3} + x_{1} x_{2} x_{4} $\\ 
$ (1, 0, 2, 0, 0) $ & $ x_{1}^{3} + x_{1}^{2} x_{2} + x_{1} x_{2}^{2} + x_{1}^{2} x_{3} + x_{1} x_{2} x_{3} + x_{1} x_{3}^{2} $\\ 
$ (1, 0, 1, 1, 0) $ & $ x_{1}^{2} x_{2} + x_{1}^{2} x_{3} + x_{1} x_{2} x_{3} + x_{1}^{2} x_{4} + x_{1} x_{2} x_{4} + x_{1} x_{3} x_{4} $\\ 
$ (0, 3, 0, 0, 0) $ & $ x_{1}^{3} + x_{1}^{2} x_{2} + x_{1} x_{2}^{2} + x_{2}^{3} $\\ 
$ (0, 2, 1, 0, 0) $ & $ x_{1}^{2} x_{2} + x_{1}^{2} x_{3} + x_{1} x_{2} x_{3} + x_{2}^{2} x_{3} $\\ 
$ (0, 2, 0, 1, 0) $ & $ x_{1}^{2} x_{2} + x_{1} x_{2}^{2} + x_{1}^{2} x_{3} + x_{1} x_{2} x_{3} + x_{2}^{2} x_{3} + x_{1}^{2} x_{4} + x_{1} x_{2} x_{4} + x_{2}^{2} x_{4} $\\ 
$ (0, 1, 2, 0, 0) $ & $ x_{1} x_{2}^{2} + x_{1} x_{2} x_{3} + x_{1} x_{3}^{2} + x_{2} x_{3}^{2} $\\ 
$ (0, 1, 1, 1, 0) $ & $ x_{1} x_{2} x_{3} + x_{1} x_{2} x_{4} + x_{1} x_{3} x_{4} + x_{2} x_{3} x_{4} $\\ 
$ (4, 0, 0, 0, 0) $ & $ x_{1}^{4} $\\ 
$ (3, 1, 0, 0, 0) $ & $ x_{1}^{3} x_{2} $\\ 
$ (3, 0, 1, 0, 0) $ & $ x_{1}^{3} x_{2} + x_{1}^{3} x_{3} $\\ 
$ (3, 0, 0, 1, 0) $ & $ x_{1}^{3} x_{2} + x_{1}^{3} x_{3} + x_{1}^{3} x_{4} $\\ 
$ (2, 2, 0, 0, 0) $ & $ x_{1}^{2} x_{2}^{2} $\\ 
$ (2, 1, 1, 0, 0) $ & $ x_{1}^{2} x_{2} x_{3} $\\ 
$ (2, 1, 0, 1, 0) $ & $ x_{1}^{2} x_{2}^{2} + x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{2} x_{4} $\\ 
$ (2, 0, 2, 0, 0) $ & $ x_{1}^{2} x_{2}^{2} + x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{3}^{2} $\\ 
$ (2, 0, 1, 1, 0) $ & $ x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{2} x_{4} + x_{1}^{2} x_{3} x_{4} $\\ 
$ (1, 3, 0, 0, 0) $ & $ x_{1} x_{2}^{3} $\\ 
$ (1, 2, 1, 0, 0) $ & $ x_{1} x_{2}^{2} x_{3} $\\ 
$ (1, 2, 0, 1, 0) $ & $ x_{1} x_{2}^{2} x_{3} + x_{1} x_{2}^{2} x_{4} $\\ 
$ (1, 1, 2, 0, 0) $ & $ x_{1} x_{2} x_{3}^{2} $\\ 
$ (1, 1, 1, 1, 0) $ & $ x_{1} x_{2} x_{3} x_{4} $\\ 
\hline
\caption{Table of forest polynomials $\forestpoly{F}$.}
\label{tab:foresttable}
\end{longtable}
\end{center}
%\pagebreak

\begin{table}
    \centering
    \begin{adjustbox}{max width=\textwidth}
    \renewcommand{\arraystretch}{1.2}
    \begin{tabular}{|c|c|c|}
    \hline
        & $\qsym{n}$  & $\sym{n}$\\
        \hline
        \textbf{Divided differences} & $\tope{i}$ &  $\partial_i$  \\
        \hline
         \textbf{Indexing combinatorics} & $F\in \indexedforests$ & $w\in S_\infty$\\
          &Fully supported forests $\suppfor{n}$ &$S_n$\\
           &Forest code $\sfc(F)$ &Lehmer code $\lcode{w}$\\
          &Left terminal set $\qdes{F}$ &Descent set $\des{w}$\\
           & $F/i$ for $i\in \qdes{F}$ &$ws_i$ for $i\in \des{w}$\\
          &Trimming sequences $\Trim{F}$ & Reduced words $\red{w}$\\
          &Zigzag forests $Z\in\zigzag{n}$ & Grassmannian permutations $\lambda$\\
         \hline
         \textbf{Monoid}&Thompson monoid &nilCoxeter monoid\\
         \hline
          \textbf{$\poly$-basis}  & Forest polynomials $\forestpoly{F}$ & Schuberts $\schub{w}$\\
         \textbf{Composites} &$\tope{F}=\tope{i_1}\cdots \tope{i_k}$ for $\textbf{i}\in \Trim{F}$ &$\partial_w=\partial_{i_1}\cdots \partial_{i_k}$ for $\textbf{i}\in \red{w}$\\
         \hline
         \textbf{$\poly_n$-basis}  & $\{\forestpoly{F}\suchthat \qdes{F}\subset [n]\}$ & $\{\schub{w}\suchthat \des{w}\subset [n]\}$\\
         \textbf{Duality} &$\ct \tope{F}\forestpoly{G}=\delta_{F,G}$ & $\ct \partial_w\schub{w'}=\delta_{w,w'}$\\
         \hline
         \textbf{Positive expansions} &$\forestpoly{F}\forestpoly{H}=\sum c^G_{F,H}\forestpoly{G}$,  $c^G_{F,H}\ge 0$ & $\schub{u}\schub{w}=\sum c^v_{u,w}\schub{v}$, $c^v_{u,w}\ge 0$\\
         \hline
        \textbf{Invariant basis} & Fundamental qsyms $\forestpoly{Z}$ & Schur polynomials $s_\lambda$\\
        \hline\textbf{Coinvariant basis}  & $\{\forestpoly{F}\suchthat F\in \suppfor{n}\}$ &$\{\schub{w}\suchthat w\in S_n\}$\\
         \textbf{Coinvariant action}&$\tope{i}:\qscoinv{n}\to \qscoinv{n-1}$ &$\partial_i:\coinv{n}\to \coinv{n}$\\
         \hline
         \textbf{Harmonic basis} & Forest volume polynomials & Degree polynomials\\
        \hline
    \end{tabular}
    \renewcommand{\arraystretch}{1}
    \end{adjustbox}
    \caption{Comparing the symmetric and quasisymmetric stories}
    \label{tab:sym_vs_qsym}
\end{table}
\printbibliography



\end{document}
