When is the chromatic quasisymmetric function symmetric?

2024/12/13 by Gillespie, Maria, Pappe, Joseph, Salois, Kyle · 2 citations
#Combinatorics (math.CO) #FOS: Mathematics #Primary: 05E05 Secondary: 05C15

paper · doi:10.48550/arxiv.2412.10556

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.

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• Chromatic quasisymmetric functions for signed graphs
• Symmetric quasisymmetric Schur-like functions

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