Colorings and positivity

Abstract

We study a version of Stanley's chromatic symmetric polynomials.

For unit interval graphs, Shareshian and Wachs introduced a refined
version with an extra parameter 'q'.

We show that many of the positivity problems regarding this family of
polynomials have analogues for unicellular LLT polynomials.

In particular, we formulate an e-positivity conjecture analogous to that
of Stanley and Stembridge.

We prove both versions of this conjecture for the path and cycle graph,
by giving an explicit e-expansion.