Chromatic symmetric functions of Dyck paths and q-rook theory (extended abstract)
arXiv:2012.00913
Abstract
The chromatic symmetric function (CSF) of Dyck paths of Stanley and its Shareshian-Wachs -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 -analogue, these results have been generalized by Abreu-Nigro (2020) and Guay-Paquet (private communication), using -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 -CSFs in a CSF basis with -hit coefficients. We further show its equivalence to the Abreu-Nigro identity expanding the -CSF in the elementary symmetric functions. This is the FPSAC extended abstract version. The full version is at ArXiv: 2104.07599.
To appear in FPSAC 2021 Proceedings. Extended abstract of ArXiv: 2104.07599