  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, 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
$q$-analogue, these results have been generalized by Abreu-Nigro (2020) 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.
