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

TL;DR

This paper provides a counterexample showing that certain trees with twenty vertices and maximum degree ten do not have Schur positive chromatic symmetric functions, disproving a previous conjecture.

Contribution

The paper presents the first known counterexample to the conjecture on Schur positivity of chromatic symmetric functions of trees.

Findings

No tree on twenty vertices with maximum degree ten has Schur positive chromatic symmetric function.

Counterexample disproves the conjecture on Schur and e-positivity of trees.

Provides new insights into the structure of chromatic symmetric functions of trees.

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