{
  "id": "2208.04113",
  "version": "v1",
  "published": "2022-08-08T13:18:17.000Z",
  "updated": "2022-08-08T13:18:17.000Z",
  "title": "Distribution of rooks on a chess-board representing a Latin square partitioned by a subsystem",
  "authors": [
    "Béla Jónás"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A $d$-dimensional generalization of a Latin square of order $n$ can be considered as a chess-board of size $n\\times n\\times \\ldots\\times n$ ($d$ times), containing $n^d$ cells with $n^{d-1}$ non-attacking rooks. Each cell is identified by a $d$-tuple $(e_1,e_2,\\ldots ,e_d)$ where $e_i \\in \\{1,2,\\ldots ,n\\}$. For $d = 3$ we prove that such a chess-board represents precisely one main class. A subsystem $T$ induced by a family of sets $<E_1,E_2,\\ldots ,E_d>$ over $\\{1,2,\\ldots ,n\\}$ is real if $E_i \\subset \\{1,2,\\ldots ,n\\}$ for each $i \\in \\{1,2,\\ldots ,d\\}$. The density of $T$ is the ratio of contained rooks to the number of cells in $T$. The distance between two subsystems is the minimum Hamming distance between cell pairs. Replacing $k$ sets of $<E_1,E_2,\\ldots ,E_d>$ by their complements, a subsystem $U$ is obtained with distance $k$ between $T$ and $U$. All these subsystems, including $T$, form a partition of the chess-board. We prove that in such a partition, the number of rooks in a $U$ and the density of $U$ can be determined from the number of rooks in $T$ and the number of cells in $T$ and $U$ and the value of $(-1)^k$. We examine the subsystem couple $(T,U)$ in the $2$- and $3$-dimensional cases, where $U$ is the most distant unique subsystem from a real $T$. On the fly, a new identity of binomial coefficients is proved.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-08T13:18:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B15"
    ],
    "keywords": [
      "latin square",
      "chess-board representing",
      "distribution",
      "distant unique subsystem",
      "main class"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}