{
  "id": "2106.14755",
  "version": "v1",
  "published": "2021-06-28T14:30:08.000Z",
  "updated": "2021-06-28T14:30:08.000Z",
  "title": "Counting Divisions of a $2\\times n$ Rectangular Grid",
  "authors": [
    "Jacob Brown"
  ],
  "comment": "9 pages, 5 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Consider a $2\\times n$ rectangular grid composed of $1\\times 1$ squares. Cutting only along the edges between squares, how many ways are there to divide the board into $k$ pieces? Building off the work of Durham and Richmond, who found the closed-form solutions for the number of divisions into 2 and 3 pieces, we prove a recursive relationship that counts the number of divisions of the board into $k$ pieces. Using this recursion, we obtain closed-form solutions for the number of divisions for $k=4$ and $k=5$ using fitting techniques on data generated from the recursion. Furthermore, we show that the closed-form solution for any fixed $k$ must be a polynomial on $n$ with degree $2k-2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-28T14:30:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A18"
    ],
    "keywords": [
      "rectangular grid",
      "counting divisions",
      "closed-form solution",
      "recursive relationship",
      "fitting techniques"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}