A counterexample to a conjecture on Schur positivity of chromatic symmetric functions of trees

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 the paper"Schur and e-positivity of trees and cut vertices".

Among the many nice results on chromatic symmetric functions in the paper [1] 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 ⌈ n 2 ⌉ has Schur positive chromatic symmetric function. In particular, Theorem 39 applies to trees. A near-converse to Theorem 39 for trees is posed in [1,Conjecture 42], which says that for every n ≥ 2, there is a tree T on n vertices, one of which has degree ⌊ n 2 ⌋, such that the chromatic symmetric function of T is Schur positive. The authors of [1] confirmed this conjecture for n ≤ 19, using computer calculations. Sadly, the conjecture is false for n = 20, as we show here. We use SageMath [2] 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 proper coloring of G is a function κ from V to the set P of positive integers such that κ(v) = κ(w) whenever {v, w} ∈ E. We fix an infinite set x := {x i : i ∈ P} of pairwise commuting variables, and write K(G) for the set of all proper colorings of G. To each proper coloring κ one associates a monomial The chromatic symmetric function X G of G is the sum of all such monomials, Chromatic symmetric functions were introduced by Stanley in [5] and have drawn considerable attention. Various results and conjectures, including the above-mentioned theorem and conjecture from [1], relate the structure of G to the expansion of X G in terms of one or more familiar bases for the algebra Λ of symmetric functions. Recall that if B is a basis for Λ and f ∈ Λ, we call f B-positive if, when we expand The Schur basis for Λ is a fundamental object in symmetric function theory. See for example [3, Chapter 7] 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 [1]. Theorem 1. If T is a tree on twenty vertices, one of which has degree ten, then X T (x) is not Schur positive.
A stable partition of G is a set partition π : V = k j=1 π j with each π j an independent set in G. We assume without loss of generality that |π j | ≥ |π j+1 | for each j ∈ [n − 1]. Setting λ j = |π j | for each j, we get that λ := (λ 1 , . . . , λ k ) is a partition of the integer |V |. We call λ the type of π. Given another partition µ = (µ 1 , . . . , µ ℓ ) of |V |, we write µ λ if λ dominates µ, that is, if m j=1 µ j ≤ m j=1 λ j for all m ∈ [k]. Our proof of Theorem 1 rests on the following basic result, due to Stanley. This result follows quickly from the fact that if µ λ, then when the Schur function s λ is expanded in the monomial basis, the coefficient of m µ is positive.
Lemma 2 (Proposition 1.5 of [4]). If X G (x) is Schur positive and G admits a stable partition of type λ, then G admits a stable partition of type µ whenever µ λ.
Corollary 3. Assume that T = (V, E) is a tree on 2n vertices and v ∈ V has degree n in T . If X T (x) is Schur positive, then every x ∈ V that is neither v nor a neighbor of v is a leaf in T .
Proof. As T is connected and bipartite, T has a unique bipartition π : V = π 1 ∪ π 2 . If X T (x) is Schur positive, then π has type (n, n) by Lemma 2. We assume without loss of generality that v ∈ π 1 . Then the neighborhood N T (v) is contained in π 2 and so π 2 = N T (v). Were the claim of the corollary false, some z ∈ V would be at distance three from v in T and therefore lie in π 2 , which is impossible.
For each partition ν = (ν 1 , . . . , ν t ) of n − 1, let T (ν) be a tree on 2n vertices in which one vertex v has exactly n neighbors v 1 , . . . , v n , and for 1 ≤ i ≤ t, v i has exactly ν i neighbors other than v (each of which is necessarily a leaf). The next result follows immediately from Corollary 3.

Corollary 4.
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 ν of n − 1 such that T is isomorphic with T (ν). Theorem 1 follows from the next result, which we prove by inspection using SageMath calculations.
Proposition 5. If ν is a partition of the integer nine, then X T (ν) is not Schur positive.