A power sum expansion for the Kromatic symmetric function

TL;DR

This paper introduces a power sum expansion formula for the Kromatic symmetric function, a K-theoretic analogue of the chromatic symmetric function, proving the integrality of coefficients and providing recursive computation methods.

Contribution

It provides an explicit formula for the ar p-expansion of ar X_G, proves the integrality of coefficients, and offers recursive methods for their calculation.

Findings

The ar p-coefficients are always integers.

The paper presents two recursive methods to compute these coefficients.

A combinatorial description and sign characterization for unweighted graphs are provided.

Abstract

The chromatic symmetric $X_G$ function is a symmetric function generalization of the chromatic polynomial of a graph, introduced by Stanley (1995). Stanley gave an expansion formula for $X_G$ in terms of the power sum symmetric functions $p_\lambda$ using the principle of inclusion-exclusion, and in arXiv:1904.01262, Bernardi and Nadeau gave an alternate $p$-expansion for $X_G$ in terms of acyclic orientations. In arXiv:2301.02177, Crew, Pechenik, and Spirkl defined the Kromatic symmetric function $\overline{X}_G$ as a $K$-theoretic analogue of $X_G$, constructed in the same way except that each vertex is assigned a nonempty set of colors such that adjacent vertices have nonoverlapping color sets. They defined a $K$-analogue $\overline{p}_\lambda$ of the power sum basis and computed the first few coefficients of the $\overline{p}$-expansion of $\overline{X}_G$ for some small graphs $G$. They conjectured that the $\overline{p}$-expansion always has integer coefficients and asked whether there is an explicit formula for these coefficients. In this note, we give a formula for the $\overline{p}$-expansion of $\overline{X}_G$, show two ways to compute the coefficients recursively (along with examples), and prove that the coefficients are indeed always integers. In a more recent paper arXiv:2502.21285, we use our formula from this note to give a combinatorial description of the $\overline{p}$-coefficients $[\overline{p}_\lambda]\overline{X}_G$ and a simple characterization of their signs in the case of unweighted graphs.