{
  "id": "0909.0683",
  "version": "v2",
  "published": "2009-09-03T16:54:31.000Z",
  "updated": "2010-04-06T07:06:19.000Z",
  "title": "A note on the total number of cycles of even and odd permutations",
  "authors": [
    "Jang Soo Kim"
  ],
  "comment": "4 pages, 2 figures, final version",
  "journal": "Discrete Math., 310 (2010) 1398-1400",
  "categories": [
    "math.CO"
  ],
  "abstract": "We prove bijectively that the total number of cycles of all even permutations of $[n]=\\{1,2,...,n\\}$ and the total number of cycles of all odd permutations of $[n]$ differ by $(-1)^n(n-2)!$, which was stated as an open problem by Mikl\\'{o}s B\\'{o}na. We also prove bijectively the following more general identity: $$\\sum_{i=1}^n c(n,i)\\cdot i \\cdot (-k)^{i-1} = (-1)^k k! (n-k-1)!,$$ where $c(n,i)$ denotes the number of permutations of $[n]$ with $i$ cycles.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-04-06T07:06:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A05",
      "05A15"
    ],
    "keywords": [
      "total number",
      "odd permutations",
      "open problem",
      "general identity"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 4,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0909.0683K"
    }
  }
}