Séminaire Lotharingien de Combinatoire, 85B.65 (2021), 12 pp.
Laura Colmenarejo, Alejandro H. Morales and Greta Panova
Chromatic Symmetric Functions of Dyck Paths and q-Rook Theory
Abstract.
The chromatic symmetric function (CSF) of Dyck paths of Stanley and its Shareshian-Wachs q-analogue have important connections to Hessenberg varieties, diagonal harmonics and LLT polynomials. In the case of, so called, abelian Dyck paths they are also curiously related to placements of non-attacking rooks by results of Stanley-Stembridge (1993) and Guay-Paquet (2013). For the q-analogue, these results have been generalized by Abreu-Nigro (2020) and Guay-Paquet (private communication), using q-hit numbers, which are a variant of the ones introduced by Garsia and Remmel. Among our main results is a new proof of Guay-Paquet's elegant identity expressing the q-CSFs in a CSF basis with q-hit coefficients. We further show its equivalence to the Abreu-Nigro identity expanding the q-CSF in the elementary symmetric functions.
Received: December 1, 2020.
Accepted: March 1, 2021.
Final version: April 29, 2021.
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