Chromatic Symmetric Functions of Hypertrees

TL;DR

This paper identifies a class of hypertrees with prime-sized edges for which the chromatic symmetric function is positive in fundamental quasisymmetric functions, providing a combinatorial interpretation for the coefficients.

Contribution

It introduces hypertrees with prime-sized edges as a class where the chromatic symmetric function is F-positive and offers an explicit combinatorial interpretation.

Findings

Chromatic symmetric function is F-positive for hypertrees with prime-sized edges.

Provides a combinatorial interpretation for F-coefficients.

Extends understanding of chromatic symmetric functions beyond ordinary graphs.

Abstract

The chromatic symmetric function $X_H$ of a hypergraph $H$ is the generating function for all colorings of $H$ so that no edge is monochromatic. When $H$ is an ordinary graph, it is known that $X_H$ is positive in the fundamental quasisymmetric functions $F_S$, but this is not the case for general hypergraphs. We exhibit a class of hypergraphs $H$ --- hypertrees with prime-sized edges --- for which $X_H$ is $F$-positive, and give an explicit combinatorial interpretation for the $F$-coefficients of $X_H$.