Séminaire Lotharingien de Combinatoire, B88c (2024), 18 pp.
Philippe Nadeau and Vasu Tewari
Down-Up Algebras and Chromatic Symmetric Functions
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We establish Guay-Paquet's unpublished linear relation between certain chromatic symmetric functions by relating his algebra on paths to the q-Klyachko algebra.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 relation 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 and 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 expansions of a combinatorial flavor. In particular we resolve a conjecture of Colmenarejo, Morales and Panova on chromatic symmetric functions. This concerns the abelian case of the Stanley-Stembridge conjecture, which we briefly survey.
Received: January 13, 2023.
Revised: April 23, 2024.
Accepted: August 13, 2024.
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