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We establish Guay-Paquet's unpublished linear relation between certain chromatic symmetric functions by relating his algebra on paths to the q-Klyachko algebra.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-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.

Remixed Eulerian numbers

“…As briefly touched upon in [18, Section 5.3], the family of remixed Eulerian numbers 𝐴 0 𝑖 𝛽0 𝑟−𝑘−𝑖 (𝑞) coincides with polynomials enumerating q-hit numbers appearing in the work of Garsia-Remmel [5]. This observation is also instrumental in order to relate these numbers to recent work around chromatic symmetric functions; see [19]. We now justify this claim.…”