{
  "id": "1711.01622",
  "version": "v1",
  "published": "2017-11-05T17:38:59.000Z",
  "updated": "2017-11-05T17:38:59.000Z",
  "title": "EW-tableaux, Le-tableaux, tree-like tableaux and the Abelian sandpile model",
  "authors": [
    "Thomas Selig",
    "Jason P. Smith",
    "Einar Steingrimsson"
  ],
  "comment": "30 pages, 8 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "A EW-tableau is a certain 0/1-filling of a Ferrers diagram, corresponding uniquely to an acyclic orientation, with a unique sink, of a certain bipartite graph called a Ferrers graph. We give a bijective proof of a result of Ehrenborg and van Willigenburg showing that EW-tableaux of a given shape are equinumerous with permutations with a given set of excedances. This leads to an explicit bijection between EW-tableaux and the much studied Le-tableaux, as well as the tree-like tableaux introduced by Aval, Boussicault and Nadeau. We show that the set of EW-tableaux on a given Ferrers diagram are in 1-1 correspondence with the minimal recurrent configurations of the Abelian sandpile model on the corresponding Ferrers graph. Another bijection between EW-tableaux and tree-like tableaux, via spanning trees on the corresponding Ferrers graphs, connects the tree-like tableaux to the minimal recurrent configurations of the Abelian sandpile model on these graphs. We introduce a variation on the EW-tableaux, which we call NEW-tableaux, and present bijections from these to Le-tableaux and tree-like tableaux. We also present results on various properties of and statistics on EW-tableaux and NEW-tableaux, as well as some open problems on these.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-05T17:38:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A19"
    ],
    "keywords": [
      "abelian sandpile model",
      "tree-like tableaux",
      "le-tableaux",
      "minimal recurrent configurations",
      "corresponding ferrers graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}