Chromatic symmetric functions of Dyck paths and q-rook theory (extended abstract)

TL;DR

This paper explores the chromatic symmetric functions of Dyck paths and their q-analogues, establishing new proofs and identities that connect these functions to rook placements and symmetric function bases.

Contribution

It provides a new proof of Guay-Paquet's identity and demonstrates the equivalence of identities involving q-CSFs, expanding understanding of their structure and connections.

Findings

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

Equivalence between q-CSF expansion in CSF basis and elementary symmetric functions

Connections established between q-analogues, rook placements, and symmetric functions

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 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.