{
  "id": "2003.03036",
  "version": "v1",
  "published": "2020-03-06T05:22:11.000Z",
  "updated": "2020-03-06T05:22:11.000Z",
  "title": "On Multitype Random Forests with a Given Degree Sequence, the Total Population of Branching Forests and Enumerations of Multitype Forests",
  "authors": [
    "Osvaldo Angtuncio Hernández"
  ],
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "In this chapter we introduce the model of multitype random forests chosen uniformly at random from the set of multitype forest with a given degree sequence, denoted by MFGDS. The unitype case was studied by Broutin and Marckert (2014). By mixing our model, one obtains multitype Galton-Watson (MGW) forests conditioned with the number of individuals by types (CMGW). The construction of MFGDS is done using the results of Chaumont and Liu (2016), and a novel path transformation on multidimensional discrete exchangeable increments processes, which is a generalization of the Vervaat transform (Vervaat 1979). We also obtain the joint law of the number of individuals by types in a MGW forest, generalizing the Otter-Dwass formula (Otter 1949, Dwass 1969). This allows us to obtain enumerations of multitype forests with a combinatorial structure (plane, labeled and binary forest), having a prescribed number of roots and individuals by types. Finally, under certain hypotheses, we give an algorithm to simulate CMGW forests, generalizing the unitype case given by Devroye (2012).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-06T05:22:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60C05",
      "05C05"
    ],
    "keywords": [
      "multitype forest",
      "degree sequence",
      "total population",
      "branching forests",
      "enumerations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}