{
  "id": "2307.13161",
  "version": "v1",
  "published": "2023-07-24T22:57:15.000Z",
  "updated": "2023-07-24T22:57:15.000Z",
  "title": "Young tableau reconstruction via minors",
  "authors": [
    "William Q. Erickson",
    "Daniel Herden",
    "Jonathan Meddaugh",
    "Mark R. Sepanski",
    "Cordell Hammon",
    "Jasmin Mohn",
    "Indalecio Ruiz-Bolanos"
  ],
  "comment": "24 pages, 18 figures",
  "categories": [
    "math.CO",
    "math.RT"
  ],
  "abstract": "The tableau reconstruction problem, posed by Monks (2009), asks the following. Starting with a standard Young tableau $T$, a 1-minor of $T$ is a tableau obtained by first deleting any cell of $T$, and then performing jeu de taquin slides to fill the resulting gap. This can be iterated to arrive at the set of $k$-minors of $T$. The problem is this: given $k$, what are the values of $n$ such that every tableau of size $n$ can be reconstructed from its set of $k$-minors? For $k=1$, the problem was recently solved by Cain and Lehtonen. In this paper, we solve the problem for $k=2$, proving the sharp lower bound $n \\geq 8$. In the case of multisets of $k$-minors, we also give a lower bound for arbitrary $k$, as a first step toward a sharp bound in the general multiset case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-24T22:57:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E10"
    ],
    "keywords": [
      "young tableau reconstruction",
      "standard young tableau",
      "general multiset case",
      "tableau reconstruction problem",
      "sharp lower bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}