{
  "id": "1709.00454",
  "version": "v1",
  "published": "2017-09-01T19:38:42.000Z",
  "updated": "2017-09-01T19:38:42.000Z",
  "title": "A directed graph generalization of chromatic quasisymmetric functions",
  "authors": [
    "Brittney Ellzey"
  ],
  "comment": "This is the full version of the FPSAC extended abstract arXiv:1612.04786",
  "categories": [
    "math.CO"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-01T19:38:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E05",
      "05A05"
    ],
    "keywords": [
      "directed graph generalization",
      "symmetric chromatic quasisymmetric functions",
      "shareshian-wachs e-positivity conjecture",
      "chromatic symmetric function",
      "circular indifference digraphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}