The lattices $\textbf m\times\textbf 2$ and $\textbf m\times\textbf 3$ are not Schur positive

Abstract

We prove that the lattices $\textbf m\times\textbf 2$ and $\textbf m\times\textbf 3$ are not Schur positive for $m\ge 8$. This confirms a conjecture of Li, Qiu, Yang, and Zhang, as an extension of counterexamples to a comment of Stanley on the universal Schur positivity of distributive lattices. Our main tools include Pieri's rules, and Wang and Wang's combinatorial formula for computing any Schur coefficient of the chromatic symmetric function of a graph in terms of special ribbon tabloids. We further show that the lattice $\textbf m\times\textbf 3$ is not strongly nice for $m\ge 44$.