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\documentclass[finalversion]{FPSAC2024}
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\usepackage{lipsum}
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\usepackage{tikz}
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\title[A signed $e$-expansion and some new $e$-positive graphs]{A signed $e$-expansion of the chromatic symmetric function and some new $e$-positive graphs}

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\author{Foster Tom\thanks{\href{mailto:ftom@mit.edu}{ftom@mit.edu}.}\addressmark{1}}

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\address{\addressmark{1}Department of Mathematics, MIT, Cambridge, MA 02139, USA\\ }

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\received{\today}

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\abstract{We prove a new signed elementary symmetric function expansion of the chromatic symmetric function of any unit interval graph. We then use sign-reversing involutions to prove new combinatorial formulas for many families of graphs, including the $K$-chains studied by Gebhard and Sagan, formed by joining cliques at single vertices, and for graphs obtained from them by removing any number of edges from any of the cut vertices. We also introduce a version for the quasisymmetric refinement of Shareshian and Wachs.}

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%\resume{\lipsum[2]}

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\keywords{chromatic symmetric function, elementary symmetric function, Stanley--Stembridge conjecture, unit interval graph}

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\usepackage[backend=bibtex]{biblatex}
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\begin{document}

\maketitle


The Stanley--Stembridge conjecture \cite{chromsym, stanstem} is one of the most actively researched open problems in algebraic combinatorics today. It asserts that if $G$ is the incomparability graph of a $(\bm 3+\bm 1)$-free poset, then $G$ is \emph{$e$-positive}, meaning that the chromatic symmetric function $X_G(\bm x)$ defined by Stanley \cite{chromsym} is a nonnegative linear combination of elementary symmetric functions. Several authors have proven that certain graphs are $e$-positive \cite{csfvertex, chrombounce3, chromqrook, trilad, lollilari, eposclasses, lltunicellschur, somepos, cyclechordepos, chrominvkostka}, studied other positivity properties of $X_G(\bm x)$, \cite{chromquasipower, eposclawcon, chromspostrees, chromposnsym, chrompos31, stanstemreduction, chromtwoparthook}, defined generalizations of the chromatic symmetric function, \cite{extendedchromsym, chromnsym, chromposquasi}, and explored implications of the Stanley--Stembridge conjecture to immanants of Jacobi--Trudi matrices \cite{stanstem}, cohomology of Hessenberg varieties \cite{chromhesssplitting, dothessenberg, chromhessequi, stanstemhess, chromquasihessenberg}, and characters of Hecke algebras \cite{chromhecke}. \\

In this extended abstract, we give a signed elementary function expansion of $X_G(\bm x)$ for any unit interval graph $G$, in terms of objects called \emph{forest triples}. We then show how sign-reversing involutions on forest triples can be used to prove combinatorial formulas for many classes of unit interval graphs, including the $K$-chains proven to be $e$-positive by Gebhard and Sagan \cite{chromnsym} and \emph{melting $K$-chains} obtained from them by removing any number of edges from any of the cut vertices. Melting $K$-chains were not previously known to be $e$-positive. We also present a generalization of our forest triple formula for the chromatic quasisymmetric function of Shareshian and Wachs \cite{chromposquasi}. 

\section{Background}

Let $G=(V,E)$ be a graph. A \emph{colouring} of $G$ is a function $\kappa:V\to\mathbb P=\{1,2,3,\ldots\}$ and we say that $\kappa$ is \emph{proper} if $\kappa(i)\neq \kappa(j)$ whenever $(i,j)\in E$. The \emph{chromatic symmetric function} of $G$ is the formal power series in infinitely many variables $\bm x=(x_1,x_2,x_3,\ldots)$ given by \cite[Definition 2.1]{chromsym}
\begin{equation}
X_G(\bm x)=\sum_{\kappa:V\to\mathbb P\text{ proper}}\bm x^\kappa,\text{ where }\bm x^\kappa=\prod_{v\in V}x_{\kappa(v)}.
\end{equation}
We are interested in expanding the symmetric function $X_G(\bm x)$ in the basis $\{e_\lambda\}$ of \emph{elementary symmetric functions} indexed by integer partitions $\lambda=\lambda_1\cdots\lambda_\ell$, defined by
\begin{equation}
e_\lambda=e_{\lambda_1}\cdots e_{\lambda_\ell},\text{ where }e_n=\sum_{i_1<\cdots<i_n}x_{i_1}\cdots x_{i_n}.
\end{equation}
We say that $G$ is \emph{$e$-positive} if the chromatic symmetric function $X_G(\bm x)$ is a nonnegative linear combination of elementary symmetric functions.

\begin{figure}\caption{\label{fig:chromsymexample} The claw graph $G$, five proper colourings of $G$, the corresponding monomials, the chromatic symmetric function $X_G(\bm x)$, the bowtie graph $H$, and the chromatic symmetric function $X_H(\bm x)$}
$$\begin{tikzpicture}
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\end{tikzpicture} \ \ 
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$$
\begin{align}
X_G(\bm x)&=\cdots+\textcolor{orange}{x_3^3}\textcolor{teal}{x_7}+\textcolor{orange}{x_3}\textcolor{teal}{x_7^3}+\cdots+\textcolor{magenta}{x_2^2}\textcolor{cyan}{x_4}\textcolor{teal}{x_7}+\textcolor{magenta}{x_2}\textcolor{cyan}{x_4^2}\textcolor{teal}{x_7}+\textcolor{magenta}{x_2}\textcolor{cyan}{x_4}\textcolor{teal}{x_7^2}+\cdots\\&\label{eq:clawe}=e_{211}-2e_{22}+5e_{31}+4e_4
\end{align}
$$\begin{tikzpicture}
\draw (1.5,0) node (1) {$H=$};
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\end{tikzpicture}$$
\begin{equation}\label{eq:bowtiee}
X_H(\bm x)=4e_{32}+12e_{41}+20e_5
\end{equation}
\end{figure}

\begin{example}
Figure \ref{fig:chromsymexample} shows the claw graph $G$. Some proper colourings of $G$, the corresponding monomials of $X_G(\bm x)$, and the $e$-expansion of $X_G(\bm x)$ are given. Because of the negative coefficient on the term $-2e_{22}$, the graph $G$ is not $e$-positive. By contrast, the bowtie graph $H$ is $e$-positive. For the complete graph $K_n$, proper colourings must use $n$ distinct colours and given $n$ distinct colours there are $n!$ proper colourings, so $X_{K_n}(\bm x)=n!e_n$ and $K_n$ is $e$-positive.
\end{example}

There has been considerable interest in characterizing $e$-positive graphs. The most prominent open problem in this direction is the Stanley--Stembridge conjecture \cite[Corollary 5.1]{chromsym}, equivalently \cite[Corollary 5.5]{stanstem}, which by a result of Guay-Paquet \cite[Theorem 5.1]{stanstemreduction} can be equivalently stated for \emph{unit interval graphs} $G$, which are graphs whose vertices can be labelled $1$ through $n$ so that
\begin{equation}\label{eq:uicondition}
\text{ for }i<j<k,\text{ if }(i,k)\in E(G),\text{ then }(i,j)\in E(G)\text{ and }(j,k)\in E(G).
\end{equation}
\begin{conjecture} (Stanley--Stembridge conjecture)
All unit interval graphs are $e$-positive. 
\end{conjecture}

\section{A signed formula}

Let $G=([n],E)$ be a \emph{natural unit interval graph}, meaning it satisfies \eqref{eq:uicondition}. We give a signed combinatorial formula for the elementary symmetric function expansion of $X_G(\bm x)$.

\begin{definition}
A subtree $T$ of $G$ is \emph{decreasing} if every vertex $v\in V(T)$ has at most one larger neighbour. A subforest $F$ of $G$ is \emph{decreasing} if all of its trees are decreasing.
\end{definition}

\begin{definition}
A \emph{tree triple} of $G$ is an object $\mathcal T=(T,\alpha,r)$ consisting of the following data.
\begin{itemize}
\item $T$ is a decreasing subtree of $G$.
\item $\alpha$ is an integer composition with size $|\alpha|=|V(T)|$.
\item $r$ is a positive integer with $1\leq r\leq\alpha_1$, the first part of $\alpha$.
\end{itemize}
A \emph{forest triple} of $G$ is a set of tree triples $\mathcal F=\{\mathcal T_i=(T_i,\alpha^{(i)},r_i)\}_{i=1}^m$ with $\sqcup_{i=1}^m V(T_i)=[n]$, so the set of trees is a decreasing spanning forest of $G$. The \emph{type} of $\mathcal F$ is the integer partition
\begin{equation}
\text{type}(\mathcal F)=\text{sort}(\alpha^{(1)}\cdots\alpha^{(m)})
\end{equation}
formed by concatenating the compositions and then sorting to form a partition. The \emph{sign} of $\mathcal F$ is the integer
\begin{equation}
\text{sign}(\mathcal F)=(-1)^{\sum_{i=1}^m(\ell(\alpha^{(i)})-1)}=(-1)^{\ell(\text{type}(\mathcal F))-m},
\end{equation}
where $\ell(\alpha)$ is the length of a composition $\alpha$. We denote by $\text{FT}(G)$ the set of forest triples of $G$ and by $\text{FT}_\mu(G)$ the set of forest triples of $G$ of type $\mu$.
\end{definition}

We now state our combinatorial formula. It was proven by first expanding $X_G(\bm x)$ in the power sum basis and then applying a change-of-basis to the elementary symmetric function basis. The technique of studying properties of $X_G(\bm x)$ by converting between different bases is explored in upcoming work of Sagan and the author \cite{chrombasis}.

\begin{theorem}\cite[Theorem 4.3]{foresttriples}
Let $G$ be a natural unit interval graph. The chromatic symmetric function $X_G(\bm x)$ satisfies
\begin{equation}\label{eq:ftformula}
X_G(\bm x)=\sum_{\mathcal F\in\text{FT}(G)}\text{sign}(\mathcal F)e_{\text{type}(\mathcal F)}=\sum_\mu\left(\sum_{\mathcal F\in\text{FT}_\mu(G)}\text{sign}(\mathcal F)\right)e_\mu.
\end{equation}
\end{theorem}

\begin{figure}\caption{\label{fig:ftexample} The bowtie graph $G$ and the forest triples of $G$ of type $32$}
$$\begin{tikzpicture}
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\draw (10.25,-4) node () {$-20e_{32}$};

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\end{figure}

\begin{example} Figure \ref{fig:ftexample} shows the forest triples of type $32$ for the bowtie graph. We can have a single tree triple $\mathcal T=(T,\alpha,r)$, in which case $\alpha$ is either $32$ or $23$ and there are either $3$ or $2$ choices for $r$. Alternatively, we can have two tree triples $\mathcal T_1=(T_1,\alpha^{(1)},r_1)$ and $\mathcal T_2=(T_2,\alpha^{(2)},r_2)$ with $\alpha^{(1)}=3=|V(T_1)|$ and $\alpha^{(2)}=2=|V(T_2)|$, and there are $3$ choices of $r_1$ and $2$ choices of $r_2$.
\end{example}

\begin{example}
For the case of $\mu=n$, forest triples $\mathcal F\in\text{FT}_n(G)$ consist of a single tree triple $\mathcal T=(T,\alpha,r)$, where $T$ is a decreasing spanning tree, we must have $\alpha=n$ so $\text{sign}(\mathcal F)=1$, and we can have any value of $1\leq r\leq n$. Because a decreasing spanning tree can be identified by specifying the unique larger neighbour of each vertex $1\leq i\leq n-1$, we have that the coefficient of $e_n$ in $X_G(\bm x)$ is $nd_1\cdots d_{n-1}$, where $d_i$ is the number of larger neighbours of vertex $i$ in $G$.
\end{example}

Now our goal is to find a sign-reversing involution on forest triples of $G$ to combinatorially prove that $G$ is $e$-positive. The structure of forest triples suggests the following approach. Let us say that a tree triple $\mathcal T=(T,\alpha,r)$ is \emph{breakable} if $\ell(\alpha)\geq 2$. In this case, we would like to somehow define a pair of forest triples $\text{break}(\mathcal T)=(\mathcal S_1,\mathcal S_2)$ of the form
\begin{equation}
\mathcal S_1=(S_1,\alpha\setminus\alpha_\ell,r)\text{ and }\mathcal S_2=(S_2,\alpha_\ell,r_2)
\end{equation}
for some decreasing trees $S_1$ and $S_2$ with $V(S_1)\sqcup V(S_2)=V(T)$ and some integer $1\leq r_2\leq\alpha_\ell$, where $\alpha_\ell$ is the last part of $\alpha$ and $\alpha\setminus\alpha_\ell$ denotes the composition with $\alpha_\ell$ removed. Let us say that the pair $(\mathcal S_1,\mathcal S_2)$ is \emph{joinable} if it is of the form $\text{break}(\mathcal T)$ for some unique $\mathcal T$, which we will denote $\text{join}(\mathcal S_1,\mathcal S_2)$. Then we would like to somehow define a map $\varphi$ on $\text{FT}(G)$ by either replacing some breakable tree triple $\mathcal T$ by $\text{break}(\mathcal T)$ or by replacing some joinable pair of tree triples $(\mathcal S_1,\mathcal S_2)$ by $\text{join}(\mathcal S_1,\mathcal S_2)$, if one exists.\\

If we can systematically choose which tree triples to replace so that $\varphi$ is an involution, then it would reverse sign because it changes the total number of tree triples by one, it would preserve type by construction, and fixed points $\mathcal F$ must have no breakable tree triples or joinable pairs of tree triples so in particular $\text{sign}(\mathcal F)=(-1)^{\sum_{i=1}^m(1-1)}=1$. Therefore, we would prove that $G$ is $e$-positive, and we would also get a combinatorial formula for the chromatic symmetric function $X_G(\bm x)$ in terms of the fixed points of $\varphi$. \\

We now demonstrate this method in the case of paths. More general results are known \cite[Section 5]{chromposquasi}, \cite[Proposition 5.3]{chromsym} but this proof technique is new. 

\begin{proposition}\label{prop:path}
The chromatic symmetric function of a path $P_n$ is given by
\begin{equation}\label{eq:pathformula}
X_{P_n}(\bm x)=\sum_{\alpha\vDash n}\alpha_1(\alpha_2-1)\cdots(\alpha_\ell-1)e_{\text{sort}(\alpha)},
\end{equation}
where the notation $\alpha\vDash n$ means that $\alpha$ is a composition with size $n$. In particular, $P_n$ is $e$-positive. 
\end{proposition} 

\begin{proof}
We label the vertices of $P_n$ so that its edges are of the form $(i,i+1)$, so decreasing subtrees of $P_n$ are paths from some $i$ to some $j>i$, which we will denote $P_{i\to j}$. For a breakable tree triple $\mathcal T=(P_{i\to j},\alpha,r)$ of $P_n$, we define $\text{break}(\mathcal T)=(\mathcal S_1,\mathcal S_2)$, where 
\begin{equation}
\mathcal S_1=(P_{i\to j-\alpha_\ell},\alpha\setminus\alpha_\ell,r)\text{ and }S_2=(P_{j-\alpha_\ell+1\to j},\alpha_\ell,1),
\end{equation} and we define a pair of tree triples $(\mathcal S_1=(P_{i\to j},\alpha^{(1)},r_1),\mathcal S_2=(P_{i'\to j'},\alpha^{(2)},r_2))$
to be \emph{joinable} if $\ell(\alpha^{(2)})=1$, $i'=j+1$, and $r_2=1$, in which case we define the tree triple \begin{equation}\text{join}(\mathcal S_1,\mathcal S_2)=(P_{i\to j'},\alpha^{(1)}\cdot\alpha^{(2)},r_1).\end{equation}
Note that $\mathcal T$ is breakable with $\text{break}(\mathcal T)=(\mathcal S_1,\mathcal S_2)$ if and only if $(\mathcal S_1,\mathcal S_2)$ is joinable with $\text{join}(\mathcal S_1,\mathcal S_2)=\mathcal T$. Now given a forest triple
\begin{equation}\mathcal F=\{\mathcal T_1=(P_{i_1\to i_2-1},\alpha^{(1)},r_1),\mathcal T_2=(P_{i_2\to i_3-1},\alpha^{(2)},r_2),\ldots,\mathcal T_\ell=(P_{i_\ell\to n},\alpha^{(\ell)},r_\ell)\},\end{equation}
we let $j$ be maximal so that either $\mathcal T_j$ is breakable or the pair $(\mathcal T_{j-1},\mathcal T_j)$ is joinable, if such a $j$ exists, and we define $\varphi(\mathcal F)$ by either breaking $\mathcal T_j$, joining $(\mathcal T_{j-1},\mathcal T_j)$, or doing nothing if no such $j$ exists. We can check that $\varphi$ is a sign-reversing involution, the fixed points can be associated with a composition $\alpha\vDash n$ by reading the tree sizes from left to right, they have type $\text{sort}(\alpha)$, and in order for no pair to be joinable, we must have $r_i\geq 2$ for every $i\geq 2$, so there are $\alpha_1(\alpha_2-1)\cdots(\alpha_\ell-1)$ choices of the $r_i$.
\end{proof}

\begin{figure}[!ht] \caption{\label{fig:pathexample} The forest triples of $P_6$ of type $222$ paired under our sign-reversing involution $\varphi$}
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\draw (3,-8) node (){FIXED};
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\end{tikzpicture}$$
\end{figure}

\begin{example} Figure \ref{fig:pathexample} shows all of the forest triples of $P_6$ of type $222$ paired under our sign-reversing involution $\varphi$. The compositions are not written but they are $2$, $22$, or $222$. We have indicated whether each $r_i$ is $1$ or $2$ by circling the $r_i$-th smallest vertex of the corresponding tree. There are $2(2-1)(2-1)=2$ fixed points, so the coefficient of $e_{222}$ in $X_{P_6}(\bm x)$ is $2$. 
\end{example}

\section{Positive formulas}

We are able to use sign-reversing involutions on forest triples to prove several combinatorial $e$-positive expansions of unit interval graphs.

\begin{definition}
Let $L_{a,b}^{(t)}$ denote the unit interval graph where a path of length $b$ is joined to a clique of size $a$, and then $t$ edges incident to the joined vertex are removed from the clique. An example is given in Figure \ref{fig:meltinglollipop}. Such graphs are called \emph{melting lollipops}. \end{definition}

Huh, Nam, and Yoo showed that melting lollipops are $e$-positive \cite[Theorem 4.9]{lltunicellschur}. A result of Aliniaeifard, Wang, and van Willigenburg \cite[Proposition 3.1]{csfvertex} implies that the sequence $(X_{L_{a,b}^{(t)}}(\bm x))_{t=0}^{a-1}$ forms an arithmetic progression. We were able to use forest triples to get a new explicit formula and see this arithmetic progression directly.

\begin{theorem}\label{thm:meltinglollipop} \cite[Theorem 5.2]{foresttriples}
The chromatic symmetric function of a melting lollipop $L_{a,b}^{(t)}$ is given by
\begin{align}\label{eq:meltinglollipop}
X_{L_{a,b}^{(t)}}(\bm x)&=t(a-2)!\sum_{\substack{\alpha\vDash n\\\alpha_\ell=a-1}}\alpha_1(\alpha_2-1)\cdots(\alpha_{\ell-1}-1)e_{\text{sort}(\alpha)}\\\nonumber&+(a-t-1)(a-2)!\sum_{\substack{\alpha\vDash n\\\alpha_\ell\geq a}}\alpha_1(\alpha_2-1)\cdots(\alpha_\ell-1)e_{\text{sort}(\alpha)}.
\end{align}
In particular, $L_{a,b}^{(t)}$ is $e$-positive.
\end{theorem}

\begin{figure}\caption{\label{fig:meltinglollipop} The melting lollipop graph $L_{6,3}^{(3)}$}
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\end{figure}

\begin{definition} 
For a composition $\gamma$ with all parts at least $2$, let $K_\gamma$ denote the unit interval graph where cliques of sizes $\gamma_1,\ldots,\gamma_\ell$ are successively joined end to end at single vertices. An example is given in Figure \ref{fig:meltingkchain}. Such graphs are called \emph{$K$-chains}.
\end{definition}

Gebhard and Sagan showed that $K$-chains are $e$-positive \cite[Corollary 7.7]{chromnsym} by using a generalization of the chromatic symmetric function in noncommuting variables. We were able to use forest triples to get a new explicit formula as a sum over a certain set $A_\gamma$ of weak compositions. 

\begin{theorem}\label{thm:kchainformula} \cite[Theorem 6.13]{foresttriples}
The chromatic symmetric function of a $K$-chain $K_\gamma$ is given by
\begin{equation}\label{eq:kchainformula}
X_{K_\gamma}(\bm x)=(\gamma_1-2)!\cdots(\gamma_{\ell-1}-2)!(\gamma_\ell-1)!\sum_{\alpha\in A_\gamma}\left(\alpha_1\prod_{i=2}^{\ell(\gamma)}|\alpha_i-(\gamma_{i-1}-1)|\right)e_{\text{sort}(\alpha)}
\end{equation}
In particular, $K_\gamma$ is $e$-positive.
\end{theorem}

\begin{example}
If $\gamma=ab$ has length $2$, we get that
\begin{equation}
X_{K_{ab}}(\bm x)=(a-1)!(b-1)!\sum_{k=\max\{a,b\}}^n(2k-n)e_{k,n-k}.
\end{equation}
\end{example}


\begin{figure}
\caption{\label{fig:meltingkchain} The $K$-chain $K_{466}$ and the melting $K$-chain $K_{466}^{(032,032)}$}
$$\begin{tikzpicture}
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(4,0)--(5.5,0.866) (4,0)--(5.5,-0.866) (6,0)--(4.5,0.866) (6,0)--(4.5,-0.866) (5.5,0.866)--(5.5,-0.866)--(4.5,0.866)--(4.5,-0.866) (5.5,0.866)--(4.5,-0.866) (6,0)--(6.5,0.866)--(7.5,0.866)--(8,0)--(7.5,-0.866)--(6.5,-0.866)--(6,0) (6,0)--(7.5,0.866)--(7.5,-0.866)--(6.5,0.866)--(6.5,-0.866)--(8,0) (7.5,0.866)--(6.5,-0.866)
(6,0)--(7.5,-0.866) (8,0)--(6.5,0.866);
\end{tikzpicture}\hspace{50pt}
\begin{tikzpicture}
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\end{tikzpicture}$$
\end{figure}

\begin{definition}
Let $\gamma$ be a composition with all parts at least $2$, and let $\epsilon$ and $\zeta$ be weak compositions with $\ell(\epsilon)=\ell(\zeta)=\ell(\gamma)$ such that $0\leq\epsilon_t,\zeta_t\leq\gamma_t-2$ for all $t$ and $\epsilon_t=0$ if and only if $\zeta_t=0$. Let $K_\gamma^{(\epsilon,\zeta)}$ denote the unit interval graph formed by removing edges from the $K$-chain $K_\gamma$ so that for all $t$, the $t$-th clique has $\epsilon_t$ edges absent from the smallest vertex and $\zeta_t$ edges absent from the largest vertex. An example is given in Figure \ref{fig:meltingkchain}. Such graphs are called \emph{melting $K$-chains}, and if every $\epsilon_t,\zeta_t\in\{0,1\}$ (so $\epsilon=\zeta$), they are called \emph{slightly melting $K$-chains}. 
\end{definition}

Aliniaeifard, Wang, and van Willigenburg showed that slightly melting $K$-chains are $e$-positive \cite[Proposition 5.5]{csfvertex}. We were able to use forest triples to get a new explicit formula as a sum over a certain set $A_\gamma^{(\epsilon)}$ of weak compositions. 

\begin{theorem}\label{thm:slightlymelting} \cite[Theorem 7.9]{foresttriples}
The chromatic symmetric function of a slightly melting $K$-chain $K_\gamma^{(\epsilon,\epsilon)}(\bm x)$ is given by
\begin{equation}\label{eq:slightlymeltingkchainformula}
X_{K_\gamma^{(\epsilon,\epsilon)}}(\bm x)=(\gamma_1-2)!\cdots(\gamma_\ell-2)!\sum_{\alpha\in A_\gamma^{\epsilon}}\left(\alpha_1\prod_{i=1}^{\ell(\gamma)}|\alpha_{i+1}-(\gamma_i-1-\epsilon_i)|\right)e_{\text{sort}(\alpha)}.
\end{equation}
In particular, $K_\gamma^{(\epsilon,\epsilon)}$ is $e$-positive.
\end{theorem}

We also proved the new result that all melting $K$-chains are $e$-positive. We have a combinatorial description of the fixed points but they are much more complicated to describe and enumerate.

\begin{theorem}
\cite[Theorem 8.3]{foresttriples}
All melting $K$-chains $K_\gamma^{(\epsilon,\zeta)}$ are $e$-positive.
\end{theorem}

It would be interesting to see whether sign-reversing involutions on forest triples could be used to show $e$-positivity of other unit interval graphs. Alternatively, we could take a dual approach where we fix $\mu$ and show that the coefficient of $e_\mu$ is nonnegative for every unit interval graph. This is done by Hwang \cite[Theorem 5.13]{chromtwoparthook} if $\mu_2=1$, by Abreu and Nigro \cite[Corollary 1.10]{chromhesssplitting} if $\ell(\mu)=2$, and in upcoming work by Sagan and the author \cite{chrombasis} if $\mu_1\leq 3$. If we can prove the following inequality, the forest triple formula would give another proof of nonnegativity for all two-part partitions.

\begin{problem}
Let $G=([n],E)$ be a natural unit interval graph and let $1\leq k\leq n-1$. Let $s_k(G)$ be the number of decreasing spanning forests $(T_1,T_2)$ of $G$, where $|V(T_2)|=k$ and $1\in V(T_1)$. Let $s(G)$ be the number of decreasing spanning trees of $G$. Prove that $ks_k(G)\geq s(G)$. 
\end{problem}

The author checked by computer that this inequality holds for all unit interval graphs $G$ with $n\leq 10$ vertices.

\section{A quasisymmetric generalization}
We also generalize our forest triple formula for the \emph{chromatic quasisymmetric function} defined by Shareshian and Wachs \cite[Definition 1.2]{chromposquasi}. 

\begin{definition}
The \emph{chromatic quasisymmetric function} of a labelled graph $G=([n],E)$ is the formal power series
\begin{equation}
X_G(\bm x;q)=\sum_{\substack{\kappa:[n]\to\mathbb P\\\kappa\text{ proper}}}q^{\text{asc}(\kappa)}\bm x^\kappa,\end{equation}
where $\text{asc}(\kappa)=|\{(i,j)\in E(G): i<j, \ \kappa(i)<\kappa(j)\}|$.
\end{definition}

Alexandersson used the following idea to study $e$-positivity of LLT polynomials \cite{lltunicellepos}.

\begin{definition}
Let $\theta\subseteq E(G)$. For a vertex $u\in [n]$, let $\text{hrv}_\theta(u)$ be the highest $v\in [n]$ reachable from $u$ by an increasing path in $([n],\theta)$ and let $\{[u_1]_\theta,\ldots,[u_m]_\theta\}$ be the set of equivalence classes of $[n]$ under the relation $u\sim_\theta u'$ if $\text{hrv}_\theta(u)=\text{hrv}_\theta(u')$. Let $\theta'\subseteq\theta$ be the subset of edges used by the increasing paths from every $u$ to $\text{hrv}_\theta(u)$ that go to the largest possible vertex at each step. We let $U(\theta)=\theta\setminus\theta'$ and the elements of $U(\theta)$ are called \emph{unnecessary edges}.
\end{definition}

\begin{figure}[!ht]
\caption{\label{fig:qversionexample} The edges of the subgraph quadruples of type $32$ for the bowtie graph $G$}

$$\begin{tikzpicture}

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\end{figure}

\begin{definition}
A \emph{subgraph quadruple} of $G$ is an object $\mathcal S=(\theta,f,\bm\alpha,\bm r)$ consisting of the following data.
\begin{itemize}
\item $\theta\subseteq E(G)$ is a subset of the edges of $G$.
\item $f:U(\theta)\to\{q,-1\}$ is a function that assigns either a $q$ or a $(-1)$ to each unnecessary edge.
\item $\bm\alpha=(\alpha^{(1)},\ldots,\alpha^{(m)})$ is a sequence of compositions such that each $|\alpha^{(i)}|=|[u_i]_\theta|$.
\item $\bm r=(r_1,\ldots,r_m)$ is a sequence of positive integers such that each $1\leq r_i\leq \alpha^{(i)}_1$.
\end{itemize}
The \emph{type} of $\mathcal S$ is the partition 
\begin{equation}
\text{type}(\mathcal S)=\text{sort}(\alpha^{(1)}\cdots\alpha^{(m)}),
\end{equation} the \emph{sign} of $\mathcal S$ is the integer
\begin{equation}
\text{sign}(\mathcal S)=(-1)^{\sum_{i=1}^m(\ell(\alpha^{(i)})-1)}(-1)^{|\{e\in U(\theta):f(e)=-1\}|},
\end{equation}
and the \emph{weight} of $\mathcal S$ is the integer
\begin{equation}
\text{weight}(\mathcal S)=\sum_{i=1}^m(r_i-1)+|\{e\in U(\theta):f(e)=q\}|.
\end{equation}
We denote by $\text{SQ}(G)$ the set of subgraph quadruples of $G$.
\end{definition}

\begin{theorem}\label{thm:qversion} \cite[Theorem 9.5]{foresttriples}
The chromatic quasisymmetric function $X_G(\bm x;q)$ of a natural unit interval graph $G$ satisfies
\begin{equation}\label{eq:qversion}
X_G(\bm x;q)=\sum_{\mathcal S\in\text{SQ}(G)}\text{sign}(\mathcal S)q^{\text{weight}(\mathcal S)}e_{\text{type}(\mathcal S)}.
\end{equation}
\end{theorem}

\begin{example}
Figure \ref{fig:qversionexample} shows the edges of the subgraph quadruples of type $32$ for the bowtie graph, where the unnecessary edges are shown by dotted lines and would each be assigned a $q$ or a $(-1)$. If we have a single equivalence class, then $\alpha$ is either $32$ or $23$ and there are either $3$ or $2$ choices for $r$. If we have two, then $\alpha^{(1)}=3$, $\alpha^{(2)}=2$, there are $3$ choices of $r_1$, and $2$ choices of $r_2$. We have written the contributions to the coefficient of $e_{32}$, taking into account the choices of $f$ and $\bm r$.
\end{example}

We can apply the earlier sign-reversing involution to subgraph quadruples to prove the following combinatorial $e$-expansion of Shareshian and Wachs \cite[Section 5]{chromposquasi}.

\begin{proposition}The chromatic quasisymmetric function of the path $P_n$ is given by
\begin{equation}
X_{P_n}(\bm x;q)=\sum_{\alpha\vDash n}q^{\ell(\alpha)-1}[\alpha_1]_q[\alpha_2-1]_q\cdots[\alpha_\ell-1]_qe_{\text{sort}(\alpha)},
\end{equation}
where the vertices of $P_n$ are labelled so that \eqref{eq:uicondition} holds and we define $[k]_q=1+q+\cdots+q^{k-1}$.
\end{proposition}

We could try to adapt our other sign-reversing involutions to subgraph quadruples.

\begin{problem}
Use subgraph quadruples to prove combinatorial $e$-positive expansions for the chromatic quasisymmetric functions $X_{K_\gamma}(\bm x;q)$ and $X_{K_{\gamma}^{(\epsilon,\zeta)}}(\bm x;q)$.
\end{problem}

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