{
  "id": "1907.07814",
  "version": "v1",
  "published": "2019-07-17T23:44:58.000Z",
  "updated": "2019-07-17T23:44:58.000Z",
  "title": "Gončarov Polynomials in Partition Lattices and Exponential Families",
  "authors": [
    "Ayomikun Adeniran",
    "Catherine Yan"
  ],
  "comment": "18 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Classical Gon\\v{c}arov polynomials arose in numerical analysis as a basis for the solutions of the Gon\\v{c}arov interpolation problem. These polynomials provide a natural algebraic tool in the enumerative theory of parking functions. By replacing the differentiation operator with a delta operator and using the theory of finite operator calculus, Lorentz, Tringali and Yan introduced the sequence of generalized Gon\\v{c}arov polynomials associated to a pair $(\\Delta, Z)$ of a delta operator $\\Delta$ and an interpolation grid $Z$. Generalized Gon\\v{c}arov polynomials share many nice algebraic properties and have a connection with the theories of binomial enumeration and order statistics. In this paper we give a complete combinatorial interpretation for any sequence of generalized Gon\\v{c}arov polynomials. First, we show that they can be realized as weight enumerators in partition lattices. Then, we give a more concrete realization in exponential families and show that these polynomials enumerate various enriched structures of vector parking functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-17T23:44:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A10",
      "05A15",
      "05A18",
      "06A07"
    ],
    "keywords": [
      "partition lattices",
      "exponential families",
      "gončarov polynomials",
      "delta operator",
      "parking functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}