{
  "id": "1306.5708",
  "version": "v3",
  "published": "2013-06-24T18:53:56.000Z",
  "updated": "2014-08-18T23:51:55.000Z",
  "title": "Blocks in cycles and k-commuting permutations",
  "authors": [
    "Rutilo Moreno",
    "Luis Manuel Rivera"
  ],
  "comment": "25 pages. v3 is a major revision",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $k$ be a nonnegative integer, and let $\\alpha$ and $\\beta$ be two permutations of $n$ symbols. We say that $\\alpha$ and $\\beta$ $k$-commute if $H(\\alpha\\beta, \\beta\\alpha)=k$, where $H$ denotes the Hamming metric between permutations. In this paper, we consider the problem of finding the permutations that $k$-commute with a given permutation. Our main result is a characterization of permutations that $k$-commute with a given permutation $\\beta$ in terms of blocks in cycles in the decomposition of $\\beta$ as a product of disjoint cycles. Using this characterization, we provide formulas for the number of permutations that $k$-commute with a transposition, a fixed-point free involution and an $n$-cycle, for any $k$. Also, we determine the number of permutations that $k$-commute with any given permutation, for $k \\leq 4$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-08-18T23:51:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A05",
      "05A15",
      "20B30"
    ],
    "keywords": [
      "k-commuting permutations",
      "fixed-point free involution",
      "disjoint cycles",
      "main result",
      "characterization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1306.5708M"
    }
  }
}