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A directed graph generalization of chromatic quasisymmetric functions
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Stanley defined the chromatic symmetric function of a graph, and Shareshian and Wachs introduced a refinement, namely the chromatic quasisymmetric function of a labeled graph. In this paper, we define the chromatic quasisymmetric function of a directed graph, which agrees with the Shareshian-Wachs definition in the acyclic case. We give an F-basis expansion for all digraphs in terms of a permutation statistic, which we call G-descents. We use this expansion to derive a p-positivity formula for all digraphs with symmetric chromatic quasisymmetric functions. We show that the chromatic quasisymmetric functions of a certain class of digraphs, called circular indifference digraphs, have symmetric coefficients. We present an e-positivity formula for the chromatic quasisymmetric function of the directed cycle, which is a t-analog of a result of Stanley. Lastly, we give a generalization of the Shareshian-Wachs e-positivity conjecture to a larger class of digraphs.
Forward citations
Cited by 2 Pith papers
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When is the chromatic quasisymmetric function symmetric?
A graph's chromatic quasisymmetric function is symmetric only under strong constraints; the paper proves connected DAGs with multiple sources/sinks are nonsymmetric and identifies a new symmetric family.
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Chromatic quasisymmetric functions for signed graphs
A new chromatic quasisymmetric invariant for directed signed graphs and an algebra SQSym of signed quasisymmetric functions are defined and studied.
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