Chromatic symmetric functions and change of basis

TL;DR

This paper investigates the appearance of elementary symmetric functions in chromatic symmetric functions, establishing conditions, connections with graph invariants, and providing explicit formulas and interpretations for certain coefficients.

Contribution

It introduces necessary conditions for elementary symmetric functions to appear in chromatic symmetric functions and relates these to graph invariants, with new formulas and interpretations.

Findings

Nonnegativity of three-column coefficients for all natural unit interval graphs

Explicit formula for the coefficient of $e_n$ in chromatic symmetric functions

New interpretation of the coefficient of $e_n$ in terms of tableaux

Abstract

We prove necessary conditions for certain elementary symmetric functions, $e_\lambda$, to appear with nonzero coefficient in Stanley's chromatic symmetric function as well as in the generalization considered by Shareshian and Wachs. We do this by first considering the expansion in the monomial or Schur basis and then performing a basis change. Using the former, we make a connection with two fundamental graph theory invariants, the independence and clique numbers. This allows us to prove nonnegativity of three-column coefficients for all natural unit interval graphs. The Schur basis permits us to give a new interpretation of the coefficient of $e_n$ in terms of tableaux. We are also able to give an explicit formula for that coefficient.