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arXiv · 2208.04175

Down-up algebras and chromatic symmetric functions

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We establish Guay-Paquet's unpublished linear relation between certain chromatic symmetric functions by relating his algebra on paths to the $q$-Klyachko algebra.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

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Philippe Nadeau, Vasu Tewari. 2022-08-08. Down-up algebras and chromatic symmetric functions. https://arxiv.org/abs/2208.04175

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