{
  "id": "2005.06456",
  "version": "v1",
  "published": "2020-05-13T17:57:56.000Z",
  "updated": "2020-05-13T17:57:56.000Z",
  "title": "Generalized Bijective Maps between $G$-Parking Functions, Spanning Trees, and the Tutte Polynomial",
  "authors": [
    "Carrie Frizzell"
  ],
  "comment": "17 pages, 9 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "We introduce an object called a tree growing sequence (TGS) in an effort to generalize bijective correspondences between $G$-parking functions, spanning trees, and the set of monomials in the Tutte polynomial of a graph $G$. A tree growing sequence determines an algorithm which can be applied to a single function, or to the set $\\mathcal{P}_{G,q}$ of $G$-parking functions. When the latter is chosen, the algorithm uses splitting operations - inspired by the recursive defintion of the Tutte polynomial - to iteratively break $\\mathcal{P}_{G,q}$ into disjoint subsets. This results in bijective maps $\\tau$ and $\\rho$ from $\\mathcal{P}_{G,q}$ to the spanning trees of $G$ and Tutte monomials, respectively. We compare the TGS algorithm to Dhar's algorithm and the family described by Chebikin and Pylyavskyy. Finally, we compute a Tutte polynomial of a zonotopal tiling using analogous splitting operations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-13T17:57:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05Cxx",
      "05C31",
      "05A19",
      "52B40",
      "05C85"
    ],
    "keywords": [
      "tutte polynomial",
      "parking functions",
      "spanning trees",
      "generalized bijective maps",
      "splitting operations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}