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 -analogue have important connections to Hessenberg varieties, diagonal harmonics and LLT polynomials. In the, so called, abelian case 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. 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. In the course of our work we establish that the -hit numbers in these expansions differ from the originally assumed Garsia-Remmel -hit numbers by certain powers of . We prove new identities for these -hit numbers, and establish connections between the three different variants.
Keywords
Cite
@article{arxiv.2104.07599,
title = {Chromatic symmetric functions of Dyck paths and q-rook theory},
author = {Laura Colmenarejo and Alejandro H. Morales and Greta Panova},
journal= {arXiv preprint arXiv:2104.07599},
year = {2023}
}
Comments
full version, 42 pages, 7 figures. arXiv admin note: text overlap with arXiv:2012.00913; v2 fixed typos, corrected proof of Prop. 6.5, updated figures and references, v3 added footnotes with updates on proofs of Conj. 6.6 and Conj. 7.1, new Prop. 6.14 and Cor. B.2. Also made a correction in Rem. 6.9