The $e$-positivity of the chromatic symmetric functions and the inverse Kostka matrix

TL;DR

This paper proves the e-positivity of certain chromatic symmetric functions associated with Dyck paths of bounce number three, using combinatorial interpretations of the inverse Kostka matrix, extending previous results beyond hook shapes.

Contribution

It introduces a new combinatorial approach to prove e-positivity for a broader class of Dyck path-related symmetric functions.

Findings

Certain coefficients in the expansion are positive

Extended e-positivity beyond hook-shape cases

Utilizes inverse Kostka matrix combinatorics

Abstract

We expand the chromatic symmetric functions for Dyck paths of bounce number three in the elementary symmetric function basis using a combinatorial interpretation of the inverse of the Kostka matrix studied in E\u{g}ecio\u{g}lu-Remmel (1990). We prove that certain coefficients in this expansion are positive. We establish the $e$-positivity of an extended class of chromatic symmetric functions for Dyck paths of bounce number three beyond the "hook-shape" case of Cho-Huh (2019).