{
  "id": "2409.05219",
  "version": "v1",
  "published": "2024-09-08T20:27:17.000Z",
  "updated": "2024-09-08T20:27:17.000Z",
  "title": "Boolean, Free, and Classical Cumulants as Tree Enumerations",
  "authors": [
    "Colin Defant",
    "Mitchell Lee"
  ],
  "comment": "18 pages, 7 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Defant found that the relationship between a sequence of (univariate) classical cumulants and the corresponding sequence of (univariate) free cumulants can be described combinatorially in terms of families of binary plane trees called troupes. Using a generalization of troupes that we call weighted troupes, we generalize this result to allow for multivariate cumulants. Our result also gives a combinatorial description of the corresponding Boolean cumulants. This allows us to answer a question of Defant regarding his troupe transform. We also provide explicit distributions whose cumulants correspond to some specific weighted troupes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-08T20:27:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A05",
      "05A18",
      "05A19",
      "46L53"
    ],
    "keywords": [
      "classical cumulants",
      "tree enumerations",
      "binary plane trees",
      "combinatorial description",
      "specific weighted troupes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}