{
  "id": "2201.13080",
  "version": "v1",
  "published": "2022-01-31T09:36:58.000Z",
  "updated": "2022-01-31T09:36:58.000Z",
  "title": "Explicit formulas for e-positivity of chromatic quasisymmetric functions",
  "authors": [
    "Seung Jin Lee",
    "Sue Kyung Y. Soh"
  ],
  "comment": "27 pages, 13 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "In 1993, Stanley and Stembridge conjectured that a chromatic symmetric function of any $(3+1)$-free poset is $e$-positive. Guay-Paquet reduced the conjecture to $(3+1)$- and $(2+2)$-free posets which are also called natural unit interval orders. Shareshian and Wachs defined chromatic quasisymmetric functions, generalizing chromatic symmetric functions, and conjectured that a chromatic quasisymmetric function of any natural unit interval order is $e$-positive and $e$-unimodal. For a given natural interval order, there is a corresponding partition $\\lambda$ and we denote the chromatic quasisymmetric function by $X_\\lambda$. The first author introduced local linear relations for chromatic quasisymmetric functions. In this paper, we prove a powerful generalization of the above-mentioned local linear relations, called a rectangular lemma, which also generalizes the result of Huh,Nam and Yoo. Such a lemma can be applied to describe explicit formulas for $e$-positivity of a chromatic symmetric function $X_\\lambda$ where $\\lambda$ is contained in a rectangle. We also suggest some conjectural formulas for $e$-positivity when $\\lambda$ is not contained in a rectangle by applying the rectangular lemma.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-01-31T09:36:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E05",
      "05A99"
    ],
    "keywords": [
      "explicit formulas",
      "chromatic symmetric function",
      "natural unit interval order",
      "local linear relations",
      "free poset"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}