{
  "id": "2107.02589",
  "version": "v1",
  "published": "2021-07-06T13:05:49.000Z",
  "updated": "2021-07-06T13:05:49.000Z",
  "title": "Connections between two classes of generalized Fibonacci numbers squared and permanents of (0,1) Toeplitz matrices",
  "authors": [
    "Michael A. Allen",
    "Kenneth Edwards"
  ],
  "comment": "10 pages, 5 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "By considering the tiling of an $N$-board (a linear array of $N$ square cells of unit width) with new types of tile that we refer to as combs, we give a combinatorial interpretation of the product of two consecutive generalized Fibonacci numbers $s_n$ (where $s_{n}=\\sum_{i=1}^q v_i s_{n-m_i}$, $s_0=1$, $s_{n<0}=0$, where $v_i$ and $m_i$ are positive integers and $m_1<\\cdots<m_q$) each raised to an arbitrary non-negative integer power. A $(w,g;m)$-comb is a tile composed of $m$ rectangular sub-tiles of dimensions $w\\times1$ separated by gaps of width $g$. The interpretation is used to give combinatorial proof of new convolution-type identities relating $s_n^2$ for the cases $q=2$, $v_i=1$, $m_1=M$, $m_2=m+1$ for $M=0,m$ to the permanent of a (0,1) Toeplitz matrix with 3 nonzero diagonals which are $-2$, $M-1$, and $m$ above the leading diagonal. When $m=1$ these identities reduce to ones connecting the Padovan and Narayana's cows numbers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-06T13:05:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A19",
      "05A05",
      "11B37",
      "11B39"
    ],
    "keywords": [
      "generalized fibonacci numbers",
      "toeplitz matrix",
      "connections",
      "narayanas cows numbers",
      "arbitrary non-negative integer power"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}