{
  "id": "math/0505217",
  "version": "v1",
  "published": "2005-05-11T14:01:44.000Z",
  "updated": "2005-05-11T14:01:44.000Z",
  "title": "The Generating Function of Ternary Trees and Continued Fractions",
  "authors": [
    "Ira Gessel",
    "Guoce Xin"
  ],
  "comment": "44 pages, 12 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Michael Somos conjectured a relation between Hankel determinants whose entries $\\frac 1{2n+1}\\binom{3n}n$ count ternary trees and the number of certain plane partitions and alternating sign matrices. Tamm evaluated these determinants by showing that the generating function for these entries has a continued fraction that is a special case of Gauss's continued fraction for a quotient of hypergeometric series. We give a systematic application of the continued fraction method to a number of similar Hankel determinants. We also describe a simple method for transforming determinants using the generating function for their entries. In this way we transform Somos's Hankel determinants to known determinants, and we obtain, up to a power of 3, a Hankel determinant for the number of alternating sign matrices. We obtain a combinatorial proof, in terms of nonintersecting paths, of determinant identities involving the number of ternary trees and more general determinant identities involving the number of $r$-ary trees.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-05-11T14:01:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15",
      "05A10",
      "05A17",
      "30B70",
      "33C05"
    ],
    "keywords": [
      "continued fraction",
      "generating function",
      "alternating sign matrices",
      "transform somoss hankel determinants",
      "count ternary trees"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......5217G"
    }
  }
}