Chromatic symmetric functions of Dyck paths and q-rook theory

TL;DR

This paper explores the chromatic symmetric functions of Dyck paths and their q-analogues, providing new proofs, identities, and connections to rook theory, Hessenberg varieties, and symmetric functions.

Contribution

It introduces a new proof of Guay-Paquet's identity, establishes equivalences between different q-hit number identities, and clarifies their relation to rook placements and symmetric functions.

Findings

New proof of Guay-Paquet's identity for q-CSFs

Equivalence between q-CSF expansions and elementary symmetric functions

Identification of differences in q-hit numbers involving powers of q

Abstract

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.