{
  "id": "2007.15932",
  "version": "v1",
  "published": "2020-07-31T09:55:49.000Z",
  "updated": "2020-07-31T09:55:49.000Z",
  "title": "Parallelogram polyominoes and rectangular EW-tableaux: correspondences through the sandpile model",
  "authors": [
    "Amal Alofi",
    "Mark Dukes"
  ],
  "comment": "18 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "This paper establishes connections between EW-tableaux and parallelogram polyominoes by using recent research regarding the sandpile model on the complete bipartite graph. This paper presents and proves a direct bijection between rectangular EW-tableaux and labelled ribbon parallelogram polyominoes. The significance of this is that allows one to move between these objects without the need for `recurrent configurations', the central object which previously tied this work together. It introduces the notion of a marked rectangular EW-tableaux that exactly encode all recurrent configurations of the sandpile model on the complete bipartite graph. This shows how non-cornersupport entries that featured in previous work can be utilized in a simple but important way in relation to EW-tableaux. It lifts the bijection between rectangular EW-tableaux and labelled ribbon parallelogram polyominoes to a bijection between marked rectangular EW-tableaux and labelled parallelogram polyominoes. This bijection helps us to fully understand the aspects of these very different objects that are, in a sense, different sides of the same coin.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-31T09:55:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sandpile model",
      "labelled ribbon parallelogram polyominoes",
      "complete bipartite graph",
      "marked rectangular ew-tableaux",
      "recurrent configurations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}