{
  "id": "2207.10252",
  "version": "v1",
  "published": "2022-07-21T01:23:06.000Z",
  "updated": "2022-07-21T01:23:06.000Z",
  "title": "Statistics of Partial Permutations via Catalan matrices",
  "authors": [
    "Yen-Jen Cheng",
    "Sen-Peng Eu",
    "Hsiang-Chun Hsu"
  ],
  "comment": "22 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "A generalized Catalan matrix $(a_{n,k})_{n,k\\ge 0}$ is generated by two seed sequences $\\mathbf{s}=(s_0,s_1,\\ldots)$ and $\\mathbf{t}=(t_1,t_2,\\ldots)$ together with a recurrence relation. By taking $s_\\ell=2\\ell+1$ and $t_\\ell=\\ell^2$ we can interpret $a_{n,k}$ as the number of partial permutations, which are $n\\times n$ $0,1$-matrices of $k$ zero rows with at most one $1$ in each row or column. In this paper we prove that most of fundamental statistics and some set-valued statistics on permutations can also be defined on partial permutations and be encoded in the seed sequences. Results on two interesting permutation families, namely the connected permutations and cycle-up-down permutations, are also given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-21T01:23:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A05",
      "05A19"
    ],
    "keywords": [
      "partial permutations",
      "seed sequences",
      "generalized catalan matrix",
      "cycle-up-down permutations",
      "recurrence relation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}