{
  "abstract": "Let $a(n)$ be the number of standard Young tableaux of shape $(n,n,n)$ whose entry in row $2$, column $1$ is odd; equivalently, the number of those whose first descent is even. This is entry A011553 of the On-Line Encyclopedia of Integer Sequences, contributed in 1996, and after thirty years of curation it carries no formula. We supply one, $a(n) = 8\\,\\bigl(n!\\,(n+2)!\\bigr)^{-1}\\sum_{m=1}^{\\lfloor n/2\\rfloor} m(m+1)(3n-2m-1)!/(n-2m)!$, and use it to settle both of the conjectures the entry records. A creative telescoping certificate shows that $a$ satisfies a linear recurrence of order two with polynomial coefficients; the order-three recurrence conjectured by R. J. Mathar in 2023 is a left multiple of it, with explicit cofactor $(4S^{-1}-3)/(7n-9)$. We also prove $a(n)\\sim 3^{3n+7/2}/(64πn^{4})$, the asymptotic conjectured by V. Kotesovec in 2014. The second proof gives slightly more than the conjecture asks: the position of the first descent has a limiting distribution, the probability that the $(2,1)$ entry equals $r+1$ tending to $r(r+2)/3^{\\,r+1}$. Summing the even terms, a uniformly random tableau of shape $(n,n,n)$ has an odd $(2,1)$ entry with probability tending to $27/64$, whereas for a uniformly random tableau of $n$ cells of unrestricted shape the corresponding limit is $1/e$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03038",
  "author_total": 1,
  "authors": [
    {
      "name": "Tong Niu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b8b526t2sbjk2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03038",
  "fulltext_url": null,
  "id": "b8b526t2sbjk2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03038",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7220394493",
  "pdf_url": "https://arxiv.org/pdf/2610.03038",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b8b526t2sbjk2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.03038",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03038"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The first descent in a standard Young tableau of shape $(n,n,n)$",
  "title_text": "The first descent in a standard Young tableau of shape (n,n,n)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/b8b526t2sbjk2",
  "venue": "arXiv",
  "versions": []
}