{
  "id": "2311.17312",
  "version": "v1",
  "published": "2023-11-29T02:09:52.000Z",
  "updated": "2023-11-29T02:09:52.000Z",
  "title": "A Correspondence between Chord Diagrams and Families of 0-1 Young Diagrams",
  "authors": [
    "Tomoki Nakamigawa"
  ],
  "comment": "11 pages, 10 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "A chord diagram is a set of chords in which no pair of chords has a common endvertex. For a chord diagram $E$ having a crossing $S = \\{ ac, bd \\}$, by the chord expansion of $E$ with respect to $S$, we have two chord diagrams $E_1 = (E\\setminus S) \\cup \\{ ab, cd \\}$ and $E_2 = (E\\setminus S) \\cup \\{ da, bc \\}$. Starting from a chord diagram $E$, by iterating expansions, we have a binary tree $T$ such that $E$ is a root of $T$ and a multiset of nonintersecting chord diagrams appear in the set of leaves of $T$. The number of leaves, which is not depending on the choice of expansions, is called the chord expansion number of $E$. A $0$-$1$ Young diagram is a Young diagram having a value of $0$ or $1$ for all boxes. This paper shows that the chord expansion number of some type counts the number of $0$-$1$ Young diagrams under some conditions. In particular, it is shown that the chord expansion number of an $n$-crossing, which corresponds to the Euler number, equals the number of $0$-$1$ Young diagrams of shape $(n,n-1,\\ldots,1)$ such that each column has at most one $1$ and each row has an even number of $1$'s.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-29T02:09:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15"
    ],
    "keywords": [
      "young diagram",
      "chord expansion number",
      "correspondence",
      "nonintersecting chord diagrams appear",
      "binary tree"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}