{
  "id": "1508.00397",
  "version": "v1",
  "published": "2015-08-03T12:26:10.000Z",
  "updated": "2015-08-03T12:26:10.000Z",
  "title": "Polyhedral geometry, supercranks, and combinatorial witnesses of congruences for partitions into three parts",
  "authors": [
    "Felix Breuer",
    "Dennis Eichhorn",
    "Brandt Kronholm"
  ],
  "comment": "28 pages, 19 figures",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "In this paper, we use a branch of polyhedral geometry, Ehrhart theory, to expand our combinatorial understanding of congruences for partition functions. Ehrhart theory allows us to give a new decomposition of partitions, which in turn allows us to define statistics called {\\it supercranks} that combinatorially witness every instance of divisibility of $p(n,3)$ by any prime $m \\equiv -1 \\pmod 6$, where $p(n,3)$ is the number of partitions of $n$ into three parts. A rearrangement of lattice points allows us to demonstrate with explicit bijections how to divide these sets of partitions into $m$ equinumerous classes. The behavior for primes $m' \\equiv 1 \\pmod 6$ is also discussed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-08-03T12:26:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "G.2.1"
    ],
    "keywords": [
      "polyhedral geometry",
      "combinatorial witnesses",
      "supercranks",
      "congruences",
      "ehrhart theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}