{
  "id": "1209.6284",
  "version": "v3",
  "published": "2012-09-27T16:53:57.000Z",
  "updated": "2014-02-25T14:18:57.000Z",
  "title": "Divisibility by 2 of Stirling numbers of the second kind and their differences",
  "authors": [
    "Jianrong Zhao",
    "Shaofang Hong",
    "Wei Zhao"
  ],
  "comment": "23 pages. To appear in Journal of Number Theory",
  "journal": "Journal of Number Theory 140 (2014), 324-348",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $n,k,a$ and $c$ be positive integers and $b$ be a nonnegative integer. Let $\\nu_2(k)$ and $s_2(k)$ be the 2-adic valuation of $k$ and the sum of binary digits of $k$, respectively. Let $S(n,k)$ be the Stirling number of the second kind. It is shown that $\\nu_2(S(c2^n,b2^{n+1}+a))\\geq s_2(a)-1,$ where $0<a<2^{n+1}$ and $2\\nmid c$. Furthermore, one gets that $\\nu_2(S(c2^{n},(c-1)2^{n}+a))=s_2(a)-1$, where $n\\geq 2$, $1\\leq a\\leq 2^n$ and $2\\nmid c$. Finally, it is proved that if $3\\leq k\\leq 2^n$ and $k$ is not a power of 2 minus 1, then $\\nu_2(S(a2^{n},k)-S(b2^{n},k))=n+\\nu_2(a-b)-\\lceil\\log_2k\\rceil +s_2(k)+\\delta(k), $ where $\\delta(4)=2$, $\\delta(k)=1$ if $k>4$ is a power of 2, and $\\delta(k)=0$ otherwise. This confirms a conjecture of Lengyel raised in 2009 except when $k$ is a power of 2 minus 1.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-02-25T14:18:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "second kind",
      "stirling number",
      "divisibility",
      "differences",
      "binary digits"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1209.6284Z"
    }
  }
}