Hall--Littlewood expansions of chromatic quasisymmetric polynomials using linked rook placements

TL;DR

This paper provides a new expansion of chromatic quasisymmetric functions into Hall--Littlewood polynomials for certain posets, using linked rook placements, and connects these to LLT polynomials through combinatorial descriptions.

Contribution

It introduces a novel Hall--Littlewood expansion for chromatic quasisymmetric functions of natural unit interval orders and links these to LLT polynomials via rook placements.

Findings

Explicit Hall--Littlewood expansion coefficients for chromatic quasisymmetric functions.

Combinatorial description of LLT polynomial coefficients in terms of rook placements.

Connection established between chromatic quasisymmetric functions and LLT polynomials.

Abstract

In this work, we obtain a Hall--Littlewood expansion of the chromatic quasisymmetric function arising from a natural unit interval order and describe the coefficients in terms of linked rook placements. Applying the Carlsson--Mellit relation between chromatic quasisymmetric functions and unicellular LLT polynomials, we also obtain a combinatorial description for the coefficients of the unicellular LLT polynomials expanded in terms of the modified transformed Hall--Littlewood polynomials.