{
  "id": "2405.07093",
  "version": "v1",
  "published": "2024-05-11T20:55:32.000Z",
  "updated": "2024-05-11T20:55:32.000Z",
  "title": "On the Correspondence Between Integer Sequences and Vacillating Tableaux",
  "authors": [
    "Zhanar Berikkyzy",
    "Pamela E. Harris",
    "Anna Pun",
    "Catherine Yan",
    "Chenchen Zhao"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A fundamental identity in the representation theory of the partition algebra is $n^k = \\sum_{\\lambda} f^\\lambda m_k^\\lambda$ for $n \\geq 2k$, where $\\lambda$ ranges over integer partitions of $n$, $f^\\lambda$ is the number of standard Young tableaux of shape $\\lambda$, and $m_k^\\lambda$ is the number of vacillating tableaux of shape $\\lambda$ and length $2k$. Using a combination of RSK insertion and jeu de taquin, Halverson and Lewandowski constructed a bijection $DI_n^k$ that maps each integer sequence in $[n]^k$ to a pair of tableaux of the same shape, where one is a standard Young tableau and the other is a vacillating tableau. In this paper, we study the fine properties of Halverson and Lewandowski's bijection and explore the correspondence between integer sequences and the vacillating tableaux via the map $DI_n^k$ for general integers $n$ and $k$. In particular, we characterize the integer sequences $\\boldsymbol{i}$ whose corresponding shape, $\\lambda$, in the image $DI_n^k(\\boldsymbol{i})$, satisfies $\\lambda_1 = n$ or $\\lambda_1 = n-k$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-11T20:55:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A05",
      "05E10"
    ],
    "keywords": [
      "integer sequence",
      "vacillating tableaux",
      "correspondence",
      "standard young tableaux",
      "partition algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}