Séminaire Lotharingien de Combinatoire, 93B.119 (2025), 12 pp.

Maria Gillespie, Joseph Pappe and Kyle Salois

When is the Chromatic Quasisymmetric Function Symmetric?

Abstract. We present several results towards the problem of determining when a chromatic quasisymmetric function (CQF) X_G(x;q) of a graph G is symmetric. We first prove the remarkable fact that if a product of two quasisymmetric functions f and g in countably infinitely many variables is symmetric, then f and g must be symmetric. This allows the problem to be reduced to the case of connected graphs.

We then show that any labeled graph having more than one source or sink has a nonsymmetric CQF. As a corollary, we find that all trees other than a directed path have a nonsymmetric CQF. We also show that a family of graphs we call "mixed mountain graphs" always have symmetric CQF.

Received: November 15, 2024. Accepted: February 15, 2025. Final version: April 1, 2025.

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