A complete multipartite basis for the chromatic symmetric function

TL;DR

This paper introduces a new basis for symmetric functions based on chromatic symmetric functions of complete multipartite graphs, providing combinatorial interpretations and connections to graph invariants.

Contribution

It establishes a complete multipartite basis for the chromatic symmetric function and interprets the change-of-basis coefficients combinatorially.

Findings

Coefficients of basis change relate to partition intersections of graph vertices.

Chromatic and Tutte symmetric functions expanded in this basis enumerate specific stable set partitions.

Provides combinatorial interpretation for basis change coefficients.

Abstract

In the vector space of symmetric functions, the elements of the basis of elementary symmetric functions are (up to a factor) the chromatic symmetric functions of disjoint unions of cliques. We consider their graph complements, the functions $\{r_{\lambda}: \lambda \text{ an integer partition}\}$ defined as chromatic symmetric functions of complete multipartite graphs. This basis was first introduced by Penaguiao [21]. We provide a combinatorial interpretation for the coefficients of the change-of-basis formula between the $r_{\lambda}$ and the monomial symmetric functions, and we show that the coefficients of the chromatic and Tutte symmetric functions of a graph $G$ when expanded in the $r$-basis enumerate certain intersections of partitions of $V(G)$ into stable sets.