H-chromatic symmetric functions

TL;DR

This paper introduces $H$-chromatic symmetric functions, generalizing Stanley's functions through $H$-colorings, and explores their properties, equivalences, and connections to classical symmetric function bases.

Contribution

It defines $H$-chromatic symmetric functions, studies their uniqueness and equivalence, and shows classical bases can be realized as such functions.

Findings

$H$-chromatic symmetric functions generalize Stanley's functions.

Identifies conditions for graph $H$-chromatic equivalence.

Classical symmetric function bases can be expressed as $H$-chromatic functions.

Abstract

We introduce $H$-chromatic symmetric functions, $X_{G}^{H}$, which use the $H$-coloring of a graph $G$ to define a generalization of Stanley's chromatic symmetric functions. We say two graphs $G_1$ and $G_2$ are $H$-chromatically equivalent if $X_{G_1}^{H} = X_{G_2}^{H}$, and use this idea to study uniqueness results for $H$-chromatic symmetric functions, with a particular emphasis on the case $H$ is a complete bipartite graph. We also show that several of the classical bases of the space of symmetric functions, i.e. the monomial symmetric functions, power sum symmetric functions, and elementary symmetric functions, can be realized as $H$-chromatic symmetric functions. We end with some conjectures and open problems.