A signed $e$-expansion of the chromatic quasisymmetric function

Abstract

We prove a new signed elementary symmetric function expansion of the chromatic quasisymmetric function of any natural unit interval graph. We then use a sign-reversing involution to prove a new combinatorial formula for K-chains, which are graphs formed by joining cliques at single vertices. This formula immediately implies $e$-positivity and $e$-unimodality for K-chains. We also prove a version of our signed $e$-expansion for arbitrary graphs.

arXiv comment36 pages, 13 figures. Replaced previous version to prove the q-analogue and simplify the proofs