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arXiv · 1709.00454

A directed graph generalization of chromatic quasisymmetric functions

Brittney Ellzey
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Abstract

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.

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Brittney Ellzey. 2017-09-05. A directed graph generalization of chromatic quasisymmetric functions. https://arxiv.org/abs/1709.00454

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