{
  "id": "0805.1273",
  "version": "v1",
  "published": "2008-05-09T03:55:53.000Z",
  "updated": "2008-05-09T03:55:53.000Z",
  "title": "Bell Polynomials and $k$-generalized Dyck Paths",
  "authors": [
    "Toufik Mansour",
    "Yidong Sun"
  ],
  "comment": "15pages, 1 figure. To appear in Discrete Applied Mathematics",
  "doi": "10.1016/j.dam.2007.10.009",
  "categories": [
    "math.CO"
  ],
  "abstract": "A {\\em k-generalized Dyck path} of length $n$ is a lattice path from $(0,0)$ to $(n,0)$ in the plane integer lattice $\\mathbb{Z}\\times\\mathbb{Z}$ consisting of horizontal-steps $(k, 0)$ for a given integer $k\\geq 0$, up-steps $(1,1)$, and down-steps $(1,-1)$, which never passes below the x-axis. The present paper studies three kinds of statistics on $k$-generalized Dyck paths: \"number of $u$-segments\", \"number of internal $u$-segments\" and \"number of $(u,h)$-segments\". The Lagrange inversion formula is used to represent the generating function for the number of $k$-generalized Dyck paths according to the statistics as a sum of the partial Bell polynomials or the potential polynomials. Many important special cases are considered leading to several surprising observations. Moreover, enumeration results related to $u$-segments and $(u,h)$-segments are also established, which produce many new combinatorial identities, and specially, two new expressions for Catalan numbers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-05-09T03:55:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A05",
      "05A15"
    ],
    "keywords": [
      "generalized dyck paths",
      "important special cases",
      "partial bell polynomials",
      "lagrange inversion formula",
      "plane integer lattice"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0805.1273M"
    }
  }
}