Down-up algebras and chromatic symmetric functions

TL;DR

This paper connects down-up algebras with chromatic symmetric functions, proving a key linear relation and resolving a conjecture related to the Stanley-Stembridge conjecture using algebraic and combinatorial methods.

Contribution

It establishes a new algebraic framework linking down-up algebras to chromatic symmetric functions and proves a conjecture in this area.

Findings

Proves Guay-Paquet's linear relation between chromatic symmetric functions.

Introduces $q$-hit polynomials as connected remixed Eulerian numbers.

Resolves a conjecture related to the Stanley-Stembridge conjecture.

Abstract

We establish Guay-Paquet's unpublished linear relation between certain chromatic symmetric functions by relating his algebra on paths to the $q$-Klyachko algebra. The coefficients in this relations are $q$-hit polynomials, and they come up naturally in our setup as connected remixed Eulerian numbers, in contrast to the computational approach of Colmenarejo-Morales-Panova. As Guay-Paquet's algebra is a down-up algebra, we are able to harness algebraic results in the context of the latter and establish results of a combinatorial flavour. In particular we resolve a conjecture of Colmenarejo-Morales-Panova on chromatic symmetric functions. This concerns the abelian case of the Stanley-Stembridge conjecture, which we briefly survey.