  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.
  This is the FPSAC extended abstract version. The full version is at ArXiv:
2104.07599.
