Chromatic symmetric functions of Dyck paths and q-rook theory

The chromatic symmetric function (CSF) of Dyck paths of Stan-ley and its Shareshian-Wachs q-analogue have important con-nections to Hessenberg varieties, diagonal harmonics and LLT polynomials. In the, so called, abelian case they are also curi-ously related to placements of non-attacking rooks by results of Stanley and Stembridge (1993) and Guay-Paquet (2013). For the q-analogue, these results have been generalized by Abreu and Nigro (2021) and Guay-Paquet (private communication), using q-hit numbers. 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. In the course of our work we establish that the q-hit numbers in these expansions differ from the originally assumed Garsia-Remmel q-hit numbers by certain powers of q. We prove new identities for these q-hit numbers, and establish connections between the three different variants.(c) 2022 Elsevier Ltd. All rights reserved.