When is the Chromatic Quasisymmetric Function Symmetric?

Gillespie, Maria; Pappe, Joseph; Salois, Kyle

doi:10.5802/alco.448

Gillespie, Maria ¹; Pappe, Joseph ¹; Salois, Kyle ²

¹ Department of Mathematics Colorado State University Fort Collins, CO 80523, USA
² Department of Mathematics, Statistics, and Computer Science St. Olaf College Northfield, MN 55057 USA

Algebraic Combinatorics, Volume 8 (2025) no. 5, pp. 1169-1192

Abstract

We investigate the problem of when a chromatic quasisymmetric function (CQF) $X_G(x;q)$ of a graph $G$ is in fact 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 in fact $f$ 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: 2024-12-29
Accepted: 2025-07-22
Published online: 2025-10-31

DOI: 10.5802/alco.448

Classification: 05E05, 05C15
Keywords: chromatic quasisymmetric function, symmetric functions, quasisymmetric functions, trees, mixed mountain graphs

Author's affiliations:

Gillespie, Maria ¹; Pappe, Joseph ¹; Salois, Kyle ²

¹ Department of Mathematics Colorado State University Fort Collins, CO 80523, USA
² Department of Mathematics, Statistics, and Computer Science St. Olaf College Northfield, MN 55057 USA

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Gillespie, Maria; Pappe, Joseph; Salois, Kyle. When is the Chromatic Quasisymmetric Function Symmetric?. Algebraic Combinatorics, Volume 8 (2025) no. 5, pp. 1169-1192. doi: 10.5802/alco.448

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