{
  "id": "math/0512249",
  "version": "v2",
  "published": "2005-12-12T21:11:50.000Z",
  "updated": "2006-02-06T20:26:05.000Z",
  "title": "A Generalization of the Ramanujan Polynomials and Plane Trees",
  "authors": [
    "Victor J. W. Guo",
    "Jiang Zeng"
  ],
  "comment": "20 pages, 2 tables, 8 figures, see also http://math.univ-lyon1.fr/~guo",
  "journal": "Adv. Appl. Math., 39 (2007), 96--115",
  "doi": "10.1016/j.aam.2006.01.004",
  "categories": [
    "math.CO"
  ],
  "abstract": "Generalizing a sequence of Lambert, Cayley and Ramanujan, Chapoton has recently introduced a polynomial sequence Q_n:=Q_n(x,y,z,t) defined by Q_1=1, Q_{n+1}=[x+nz+(y+t)(n+y\\partial_y)]Q_n. In this paper we prove Chapoton's conjecture on the duality formula: Q_n(x,y,z,t)=Q_n(x+nz+nt,y,-t,-z), and answer his question about the combinatorial interpretation of Q_n. Actually we give combinatorial interpretations of these polynomials in terms of plane trees, half-mobile trees, and forests of plane trees. Our approach also leads to a general formula that unifies several known results for enumerating trees and plane trees.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-02-06T20:26:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15",
      "05C05"
    ],
    "keywords": [
      "plane trees",
      "ramanujan polynomials",
      "generalization",
      "combinatorial interpretation",
      "duality formula"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math.....12249G"
    }
  }
}