Combinatorial Theory

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 the chromatic symmetric function for arbitrary graphs.

Mathematics Subject Classifications: 05E05, 05E10, 05C15

Keywords: Chromatic quasisymmetric function, elementary symmetric function, natural unit interval graph, proper colouring, Shareshian-Wachs conjecture, Stanley-Stembridge conjecture