{
  "id": "2105.09000",
  "version": "v1",
  "published": "2021-05-19T09:09:09.000Z",
  "updated": "2021-05-19T09:09:09.000Z",
  "title": "Continuants with equal values, a combinatorial approach",
  "authors": [
    "Gerhard Ramharter",
    "Luca Q. Zamboni"
  ],
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "A regular continuant is the denominator $K$ of a terminating regular continued fraction, interpreted as a function of the partial quotients. We regard $K$ as a function defined on the set of all finite words on the alphabet $1<2<3<\\dots$ with values in the positive integers. Given a word $w=w_1\\cdots w_n$ with $w_i\\in\\mathbb{N}$ we define its multiplicity $\\mu(w)$ as the number of times the value $K(w)$ is assumed in the Abelian class $\\mathcal{X}(w)$ of all permutations of the word $w.$ We prove that there is an infinity of different lacunary alphabets of the form $\\{b_1<\\dots <b_t<l+1<l+2<\\dots <s\\}$ with $b_j, t, l, s\\in\\mathbb{N}$ and $s$ sufficiently large such that $\\mu$ takes arbitrarily large values for words on these alphabets. The method of proof relies in part on a combinatorial characterisation of the word $w_{max}$ in the class $\\mathcal{X}(w)$ where $K$ assumes its maximum.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-19T09:09:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J70",
      "68R15",
      "68W05",
      "05A20"
    ],
    "keywords": [
      "combinatorial approach",
      "equal values",
      "abelian class",
      "finite words",
      "terminating regular continued fraction"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}