A directed graph generalization of chromatic quasisymmetric functions

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.

arXiv commentThis is the full version of the FPSAC extended abstract arXiv:1612.04786

References in corpus (4)

• A modular relation for the chromatic symmetric functions of (3+1)-free posets
• LLT polynomials, chromatic quasisymmetric functions and graphs with cycles
• Chromatic quasisymmetric functions of directed graphs
• Symmetric function generalizations of graph polynomials

Cited by in corpus (5)

• LLT polynomials, chromatic quasisymmetric functions and graphs with cycles
• Chromatic quasisymmetric functions of directed graphs
• A combinatorial expansion of vertical-strip LLT polynomials in the basis of elementary symmetric functions
• Modular Relations of the Tutte Symmetric Function
• Reconstructing Rooted Trees From Their Strict Order Quasisymmetric Functions