{
  "id": "math/9804119",
  "version": "v2",
  "published": "1998-04-24T16:32:41.000Z",
  "updated": "1999-01-28T22:25:20.000Z",
  "title": "Enumeration of m-ary cacti",
  "authors": [
    "Miklos Bona",
    "Michel Bousquet",
    "Gilbert Labelle",
    "Pierre Leroux"
  ],
  "comment": "LaTeX2e, 28 pages, 9 figures (eps), 3 tables",
  "journal": "Advances in Applied Mathematics, 24 (2000), 22-56",
  "categories": [
    "math.CO"
  ],
  "abstract": "The purpose of this paper is to enumerate various classes of cyclically colored m-gonal plane cacti, called m-ary cacti. This combinatorial problem is motivated by the topological classification of complex polynomials having at most m critical values, studied by Zvonkin and others. We obtain explicit formulae for both labelled and unlabelled m-ary cacti, according to i) the number of polygons, ii) the vertex-color distribution, iii) the vertex-degree distribution of each color. We also enumerate m-ary cacti according to the order of their automorphism group. Using a generalization of Otter's formula, we express the species of m-ary cacti in terms of rooted and of pointed cacti. A variant of the m-dimensional Lagrange inversion is then used to enumerate these structures. The method of Liskovets for the enumeration of unrooted planar maps can also be adapted to m-ary cacti.",
  "revisions": [
    {
      "version": "v2",
      "updated": "1999-01-28T22:25:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15",
      "05C30"
    ],
    "keywords": [
      "enumeration",
      "cyclically colored m-gonal plane cacti",
      "enumerate m-ary cacti",
      "m-dimensional lagrange inversion",
      "complex polynomials"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1998math......4119B"
    }
  }
}