{
  "id": "2303.15756",
  "version": "v1",
  "published": "2023-03-28T06:32:17.000Z",
  "updated": "2023-03-28T06:32:17.000Z",
  "title": "Complete non-ambiguous trees and associated permutations: connections through the Abelian sandpile model",
  "authors": [
    "Thomas Selig",
    "Haoyue Zhu"
  ],
  "comment": "29 pages, 15 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "We study a link between complete non-ambiguous trees (CNATs) and permutations exhibited by Daniel Chen and Sebastian Ohlig in recent work. In this, they associate a certain permutation to the leaves of a CNAT, and show that the number of $n$-permutations that are associated with exactly one CNAT is $2^{n-2}$. We connect this to work by the first author and co-authors linking complete non-ambiguous trees and the Abelian sandpile model. This allows us to prove a number of conjectures by Chen and Ohlig on the number of $n$-permutations that are associated with exactly $k$ CNATs for various $k > 1$, via bijective correspondences between such permutations. We also exhibit a new bijection between $(n-1)$-permutations and CNATs whose permutation is the decreasing permutation $n(n-1)\\cdots1$. This bijection maps the left-to-right minima of the permutation to dots on the bottom row of the corresponding CNAT, and descents of the permutation to empty rows of the CNAT.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-03-28T06:32:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A19",
      "05A05",
      "05A15",
      "05C30"
    ],
    "keywords": [
      "abelian sandpile model",
      "associated permutations",
      "connections",
      "co-authors linking complete non-ambiguous trees",
      "empty rows"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}