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

TL;DR

This paper introduces a new signed elementary symmetric function expansion for the chromatic quasisymmetric function of certain graphs, providing combinatorial formulas and positivity results.

Contribution

It presents a novel signed $e$-expansion for the chromatic quasisymmetric function and a combinatorial formula for K-chains, extending to arbitrary graphs.

Findings

Proves $e$-positivity and $e$-unimodality for K-chains

Develops a sign-reversing involution for combinatorial formulas

Extends signed $e$-expansion to arbitrary graphs

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.