{
  "id": "2408.06923",
  "version": "v1",
  "published": "2024-08-13T14:15:22.000Z",
  "updated": "2024-08-13T14:15:22.000Z",
  "title": "Skeletal generalizations of Dyck paths, parking functions, and chip-firing games",
  "authors": [
    "Spencer Backman",
    "Cole Charbonneau",
    "Nicholas A. Loehr",
    "Patrick Mullins",
    "Mazie O'Connor",
    "Gregory S. Warrington"
  ],
  "comment": "29 pages, 9 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "For $0\\leq k\\leq n-1$, we introduce a family of $k$-skeletal paths which are counted by the $n$-th Catalan number for each $k$, and specialize to Dyck paths when $k=n-1$. We similarly introduce $k$-skeletal parking functions which are equinumerous with the spanning trees on $n+1$ vertices for each $k$, and specialize to classical parking functions for $k=n-1$. The preceding constructions are generalized to paths lying in a trapezoid with base $c > 0$ and southeastern diagonal of slope $1/m$; $c$ and $m$ need not be integers. We give bijections among these families when $k$ varies with $m$ and $c$ fixed. Our constructions are motivated by chip firing and have connections to combinatorial representation theory and tropical geometry.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-13T14:15:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15",
      "05A19",
      "05C57"
    ],
    "keywords": [
      "dyck paths",
      "skeletal generalizations",
      "chip-firing games",
      "th catalan number",
      "combinatorial representation theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}