Abstract
The chromatic quasisymmetric function of a graph was introduced by Shareshian and Wachs as a refinement of Stanley's chromatic symmetric function. An explicit combinatorial formula, conjectured by Shareshian and Wachs, expressing the chromatic quasisymmetric function of the incomparability graph of a natural unit interval order in terms of power sum symmetric functions, is proven. The proof uses a formula of Roichman for the irreducible characters of the symmetric group.
Highlights
The chromatic quasisymmetric function of a graph was introduced by Shareshian and Wachs as a refinement of Stanley’s chromatic symmetric function
The chromatic quasisymmetric function of a graph was introduced by Shareshian and Wachs [3, 4] as a refinement of Stanley’s symmetric function generalization of the chromatic polynomial of a graph [5]
Given a sequence x = (x1, x2, . . . ) of commuting indeterminates, the chromatic quasisymmetric function of G is defined as XG(x, t) =
Summary
The chromatic quasisymmetric function of a graph was introduced by Shareshian and Wachs [3, 4] as a refinement of Stanley’s symmetric function generalization of the chromatic polynomial of a graph [5]. The function XG is quasisymmetric in x; its study [3, 4] connects seemingly disparate topics, such as graph colorings, permutation statistics and the cohomology of Hessenberg varieties, and provides valuable insight into Stanley’s chromatic symmetric function [5], to which XG reduces for t = 1. The present note confirms one of the conjectures of [3, 4] (see [3, Conjecture 4.15] [4, Conjecture 7.6]), predicting an explicit combinatorial formula which expresses XG in terms of power sum symmetric functions This formula refines a formula of Stanley [5, Theorem 2.6] and provides some evidence in favor of the positivity conjecture of Shareshian and Wachs (see [3, Conjecture 4.9] [4, Conjecture 1.3]) on the expansion of XG in the basis of elementary symmetric functions.
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