On Macdonald expansions of 𝑞-chromatic symmetric functions and the Stanley–Stembridge conjecture

Abstract

The Stanley–Stembridge conjecture asserts that the chromatic symmetric function of the incomparability graph of a

-free poset is 𝑒-positive. Recently, Hikita proved this conjecture by giving an explicit 𝑒-expansion of the Shareshian–Wachs 𝑞-chromatic refinement for unit interval graphs. Using the

A q , t \mathbb{A}_{q,t}

algebra, we give an expansion of these 𝑞-chromatic symmetric functions into Macdonald polynomials. Upon setting

t = 1 t=1

, we obtain another proof of the Stanley–Stembridge conjecture and re-derive Hikita’s formula. Upon setting

t = 0 t=0

, we obtain an expansion into Hall–Littlewood symmetric functions.