{
  "id": "2007.15562",
  "version": "v1",
  "published": "2020-07-30T16:11:27.000Z",
  "updated": "2020-07-30T16:11:27.000Z",
  "title": "Down-step statistics in generalized Dyck paths",
  "authors": [
    "Andrei Asinowski",
    "Benjamin Hackl",
    "Sarah J. Selkirk"
  ],
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "The number of down-steps between pairs of up-steps in $k_t$-Dyck paths, a recently introduced generalization of Dyck paths consisting of steps $\\{(1, k), (1, -1)\\}$ such that the path stays (weakly) above the line $y=-t$, is studied. Results are proved bijectively and by means of generating functions, and lead to several interesting identities as well as links to other combinatorial structures. In particular, there is a connection between $k_t$-Dyck paths and perforation patterns for punctured convolutional codes (binary matrices) used in coding theory. Surprisingly, upon restriction to usual Dyck paths this yields a new combinatorial interpretation of Catalan numbers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-30T16:11:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15",
      "05A19",
      "05A05"
    ],
    "keywords": [
      "generalized dyck paths",
      "down-step statistics",
      "usual dyck paths",
      "catalan numbers",
      "binary matrices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}