Séminaire Lotharingien de Combinatoire, 95B.79 (2026), 12 pp.

Sean T. Griffin, Anton Mellit, Marino Romero, Kevin Weigl and Joshua Jeishing Wen

A Macdonald Expansion of the 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 (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 algebra, we give an
expansion of these q-chromatic symmetric functions into
Macdonald polynomials. Since q-chromatic symmetric functions
do not depend on t, we obtain expansions into different bases
of symmetric functions by specializing t. Upon
setting t = 1, we rederive Hikita's formula and obtain
another proof of the Stanley-Stembridge Conjecture. Upon
setting t = 0, we obtain an expansion into Hall-Littlewood
symmetric functions.

Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.

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