Chromatic symmetric functions of Dyck paths and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e674" altimg="si192.svg"><mml:mi>q</mml:mi></mml:math>-rook theory

The chromatic symmetric function (CSF) of Dyck paths Stanley and its Shareshian–Wachs q-analogue have important connections to Hessenberg varieties, diagonal harmonics LLT polynomials. In the, so called, abelian case they are also curiously related placements non-attacking rooks by results Stembridge (1993) Guay-Paquet (2013). For the q-analogue, these been generalized Abreu Nigro (2021) (private communication), using q-hit numbers. Among our main is a new proof Guay-Paquet’s elegant identity expressing q-CSFs in CSF basis with coefficients. We further show equivalence Abreu–Nigro expanding q-CSF elementary functions. course work we establish that numbers expansions differ from originally assumed Garsia–Remmel certain powers q. prove identities for numbers, between three different variants.