2022
DOI: 10.48550/arxiv.2208.04175
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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Abstract: We establish Guay-Paquet's unpublished linear relation between certain chromatic symmetric functions by relating his algebra on paths to the q-Klyachko algebra.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 c… Show more

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“…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.…”
Section: Interval Support and Q-hit Numbersmentioning
confidence: 52%
“…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.…”
Section: Interval Support and Q-hit Numbersmentioning
confidence: 52%