{
  "id": "0805.4358",
  "version": "v1",
  "published": "2008-05-28T14:20:28.000Z",
  "updated": "2008-05-28T14:20:28.000Z",
  "title": "Potential Polynomials and Motzkin Paths",
  "authors": [
    "Yidong Sun"
  ],
  "comment": "11 pages, 1 figures; Discreste Math., to appear",
  "categories": [
    "math.CO"
  ],
  "abstract": "A {\\em Motzkin 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 $(1, 0)$, up-steps $(1,1)$, and down-steps $(1,-1)$, which never passes below the x-axis. A {\\em $u$-segment {\\rm (resp.} $h$-segment {\\rm)}} of a Motzkin path is a maximum sequence of consecutive up-steps ({\\rm resp.} horizontal-steps). The present paper studies two kinds of statistics on Motzkin paths: \"number of $u$-segments\" and \"number of $h$-segments\". The Lagrange inversion formula is utilized to represent the weighted generating function for the number of Motzkin paths according to the statistics as a sum of the partial Bell polynomials or the potential polynomials. As an application, a general framework for studying compositions are also provided.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-05-28T14:20:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A05",
      "05A15"
    ],
    "keywords": [
      "motzkin path",
      "potential polynomials",
      "partial bell polynomials",
      "lagrange inversion formula",
      "plane integer lattice"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0805.4358S"
    }
  }
}