{
  "id": "1708.00513",
  "version": "v1",
  "published": "2017-08-01T21:06:15.000Z",
  "updated": "2017-08-01T21:06:15.000Z",
  "title": "From Dyck paths to standard Young tableaux",
  "authors": [
    "Juan B. Gil",
    "Peter R. W. McNamara",
    "Jordan O. Tirrell",
    "Michael D. Weiner"
  ],
  "comment": "19 pages, 9 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "The number of Dyck paths of semilength $n$ is certainly not equal to the number of standard Young tableaux (SYT) with $n$ boxes. We investigate several ways to add structure or restrict these sets so as to obtain equinumerous sets. Our most sophisticated bijective proof starts with Dyck paths whose $k$-ascents for $k>1$ are labeled by connected matchings on $[k]$ and arrives at SYT with at most $2k-1$ rows. Along the way, this bijection visits $k$-noncrossing and $k$-nonnesting partial matchings, oscillating tableaux and involutions with decreasing subsequences of length at most $2k-1$. In addition, we present bijections from eight other types of Dyck and Motzkin paths to certain classes of SYT.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-01T21:06:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A19"
    ],
    "keywords": [
      "standard young tableaux",
      "dyck paths",
      "motzkin paths",
      "nonnesting partial matchings",
      "bijection visits"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}