A quasisymmetric function generalization of the chromatic symmetric function

TL;DR

This paper introduces a new quasisymmetric generalization of the chromatic symmetric function for graphs, extending Stanley's work and revealing positivity properties and polynomial evaluations that relate to acyclic orientations.

Contribution

It defines the $k$-chromatic quasisymmetric function $X^k_G$, proves its positivity in the fundamental basis, and generalizes Stanley's theorem on acyclic orientations via a new polynomial $ ext{}\chi^k_G( ext{)}$.

Findings

$X^k_G$ is positive in the fundamental basis.

Evaluations of $ ext{}\chi^k_G( ext{)}$ at negative values generalize Stanley's theorem.

The paper establishes a connection between the new polynomial and acyclic orientations.

Abstract

The chromatic symmetric function $X_G$ of a graph $G$ was introduced by Stanley. In this paper we introduce a quasisymmetric generalization $X^k_G$ called the $k$-chromatic quasisymmetric function of $G$ and show that it is positive in the fundamental basis for the quasisymmetric functions. Following the specialization of $X_G$ to $\chi_G(\lambda)$, the chromatic polynomial, we also define a generalization $\chi^k_G(\lambda)$ and show that evaluations of this polynomial for negative values generalize a theorem of Stanley relating acyclic orientations to the chromatic polynomial.