Chromatic quasisymmetric functions

TL;DR

This paper introduces a refined version of Stanley's chromatic symmetric function called the chromatic quasisymmetric function, exploring its properties, expansions, conjectures, and connections to algebraic and geometric structures.

Contribution

It develops a quasisymmetric refinement of the chromatic symmetric function, derives new basis expansions, and proposes conjectures linking it to representation theory and combinatorial positivity.

Findings

Refined Gasharov's Schur-basis expansion.

Derived Chow's expansion in fundamental quasisymmetric functions.

Proved special cases of a conjectural power sum expansion.

Abstract

We introduce a quasisymmetric refinement of Stanley's chromatic symmetric function. We derive refinements of both Gasharov's Schur-basis expansion of the chromatic symmetric function and Chow's expansion in Gessel's basis of fundamental quasisymmetric functions. We present a conjectural refinement of Stanley's power sum basis expansion, which we prove in special cases. We describe connections between the chromatic quasisymmetric function and both the $q$-Eulerian polynomials introduced in our earlier work and, conjecturally, representations of symmetric groups on cohomology of regular semisimple Hessenberg varieties, which have been studied by Tymoczko and others. We discuss an approach, using the results and conjectures herein, to the $e$-positivity conjecture of Stanley and Stembridge for incomparability graphs of $(3+1)$-free posets.