\begin{document}

\title[Schur positivity of trees]{A counterexample to a conjecture on Schur positivity of chromatic symmetric functions of trees}

\author{Emmanuella Sandratra Rambeloson}
\address{African Institute of Mathematical Sciences \\ Accra, Ghana}
\email{emmanuella@aims.ac.za}

\author{John Shareshian}
\address{Washington University \\ St Louis, MO, USA}
\email{jshareshian@wustl.edu}

\maketitle

\begin{abstract}
We show that no tree on twenty vertices with maximum degree ten has Schur positive chromatic symmetric function, thereby providing a counterexample to a conjecture from \cite{DSW}.
\end{abstract}

Among the many nice results on chromatic symmetric functions in the paper \cite{DSW} of Dahlberg, She and van Willigenburg is Theorem 39 therein, which says that no bipartite graph on $n$ vertices with a vertex of degree more than $\lceil \frac{n}{2} \rceil$ has Schur positive chromatic symmetric function.  In particular, Theorem 39 applies to trees.  A near-converse to Theorem 39 for trees is posed in \cite[Conjecture 42]{DSW}, which says that for every $n \geq 2$, there is a tree $T$ on $n$ vertices, one of which has degree $\lfloor \frac{n}{2} \rfloor$, such that the chromatic symmetric function of $T$ is Schur positive.  The authors of \cite{DSW} confirmed this conjecture for $n \leq 19$, using computer calculations.  Sadly, the conjecture is false for $n=20$, as we show here. We use SageMath \cite{SAGE} calculations after a preparatory proposition that reduces the number of trees that we must examine.


We give the requisite definitions and reiterate more formally.  Given a (finite, loopless, simple) graph $G=(V,E)$, a {\sf proper coloring} of $G$ is a function $\kappa$ from $V$ to the set $\P$ of positive integers such that $\kappa(v) \neq \kappa(w)$ whenever $\{v,w\} \in E$.  We fix an infinite set $\x:=\{x_i:i \in \P\}$ of pairwise commuting variables, and write $\K(G)$ for the set of all proper colorings of $G$. To each proper coloring $\kappa$ one associates a monomial 
$$
\x^\kappa:=\prod_{v \in V}x_{\kappa(v)}.
$$
The {\sf chromatic symmetric function} $X_G$ of $G$ is the sum of all such monomials,
$$
X_G(\x):=\sum_{\kappa \in \K(G)}\x^\kappa.
$$
Chromatic symmetric functions were introduced by Stanley in \cite{StCSF} and have drawn considerable attention.  Various results and conjectures, including the above-mentioned theorem and conjecture from \cite{DSW}, relate the structure of $G$ to the expansion of $X_G$ in terms of one or more familiar bases for the algebra $\Lambda$ of symmetric functions.   Recall that if $B$ is a basis for $\Lambda$ and $f \in \Lambda$, we call $f$ {\sf $B$-positive} if, when we expand $f=\sum_{b \in B}\alpha_b b$, each $\alpha_b$ is non-negative.  The {\sf Schur basis} for $\Lambda$ is a fundamental object in symmetric function theory. See for example \cite[Chapter 7]{StEC} for basic properties of Schur functions and other rudimentary facts about symmetric functions that will be used herein without reference.


We prove the following result, thereby disproving Conjecture 42 of \cite{DSW}.

\begin{thm} \label{main}
If $T$ is a tree on twenty vertices, one of which has degree ten, then $X_T(\x)$ is not Schur positive.
\end{thm}

A {\sf stable partition} of $G$ is a set partition $\pi:V=\bigcup_{j=1}^k \pi_j$ with each $\pi_j$ an independent set in $G$.  We assume without loss of generality that $|\pi_j| \geq |\pi_{j+1}|$ for each $j \in [n-1]$.   Setting $\lambda_j=|\pi_j|$ for each $j$, we get that $\lambda:=(\lambda_1,\ldots,\lambda_k)$ is a partition of the integer $|V|$. We call $\lambda$ the {\sf type} of $\pi$.  Given another partition $\mu=(\mu_1,\ldots,\mu_\ell)$ of $|V|$, we write $\mu \preceq \lambda$ if $\lambda$ {\sf dominates} $\mu$, that is, if $\sum_{j=1}^m \mu_j \leq \sum_{j=1}^m \lambda_j$ for all $m \in [k]$.  Our proof of Theorem \ref{main} rests on the following basic result, due to Stanley.  This result follows quickly from the fact that if $\mu \preceq \lambda$, then when the Schur function $s_\lambda$ is expanded in the monomial basis, the coefficient of $m_\mu$ is positive.
\begin{lemma}[Proposition 1.5 of \cite{StCSF2}] \label{domlem}
If $X_G(\x)$ is Schur positive and $G$ admits a stable partition of type $\lambda$, then $G$ admits a stable partition of type $\mu$ whenever $\mu \preceq \lambda$.
\end{lemma}

\begin{cor} \label{maincor}
Assume that $T=(V,E)$ is a tree on $2n$ vertices and $v \in V$ has degree $n$ in $T$.  If $X_T(\x)$ is Schur positive, then every $x \in V$ that is neither $v$ nor a neighbor of $v$ is a leaf in $T$.
\end{cor}

\begin{proof}
As $T$ is connected and bipartite, $T$ has a unique bipartition $\pi:V=\pi_1 \cup \pi_2$.  If $X_T(\x)$ is Schur positive, then $\pi$ has type $(n,n)$ by Lemma \ref{domlem}.  We assume without loss of generality that $v \in \pi_1$.  Then the neighborhood $N_T(v)$ is contained in $\pi_2$ and so $\pi_2=N_T(v)$.  Were the claim of the corollary false, some $z \in V$ would be at distance three from $v$ in $T$ and therefore lie in $\pi_2$, which is impossible.     
\end{proof}

For each partition $\nu=(\nu_1,\ldots,\nu_t)$ of $n-1$, let $T(\nu)$ be a tree on $2n$ vertices in which one vertex $v$ has exactly $n$ neighbors $v_1,\ldots,v_n$, and for $1 \leq i \leq t$, $v_i$ has exactly $\nu_i$ neighbors other than $v$ (each of which is necessarily a leaf).  The next result follows immediately from Corollary \ref{maincor}.

\begin{cor} \label{maincor2}
If $T$ is a tree on $2n$ vertices, one of which has degree $n$, and $X_T(\x)$ is Schur positive, then there is some partition $\nu$ of $n-1$ such that $T$ is isomorphic with $T(\nu)$.
\end{cor}

Theorem \ref{main} follows from the next result, which we prove by inspection using SageMath calculations.

\begin{prop} \label{lastprop}
If $\nu$ is a partition of the integer nine, then $X_{T(\nu)}$ is not Schur positive.
\end{prop}

Our computations reveal in particular that if $n=10$ and $\nu_1 \geq 6$, then the coefficient of $s_{(9,9,2)}$ in the Schur expansion of $X_{T(\nu)}(\x)$ is negative; and if $n=10$ and $\nu_1 \leq 5$, then the coefficient of $s_{(3,3,2,2,2,2,2,2,2)}$ in the Schur expansion of $X_{T(\nu)}(\x)$ is negative.  This Schur expansion has can have as few as four negative coefficients (when $\nu$ is one of $(6,2,1)$, $(6,1,1,1)$ or $(5,4)$) and as many as thirty (when $\nu$ is one of $(2,2,2,2,1)$, $(2,2,2,1,1,1)$ or $(1,1,1,1,1,1,1,1,1)$).


Our programs, along with the complete Schur expansion of $X_{T(\nu)}(\x)$ for each partition $\nu$ of nine, can be found at \url{https://github.com/emmanuellasa/Schur_Decomposition_20}.


\section*{Acknowledgement} We thank Stephanie van Willigenburg for helpful comments.
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