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

Abstract

Abstract The Stanley–Stembridge conjecture asserts that the chromatic symmetric function of the incomparability graph of a ( 3 + 1 ) (3+1) -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.

The authors' abstract, as published at the source. Journal für die reine und angewandte Mathematik (Crelles Journal), 2026 · DOI ↗