{
  "id": "1908.05594",
  "version": "v1",
  "published": "2019-08-15T15:44:10.000Z",
  "updated": "2019-08-15T15:44:10.000Z",
  "title": "On the $p$-adic properties of Stirling numbers of the first kind",
  "authors": [
    "Shaofang Hong",
    "Min Qiu"
  ],
  "comment": "22 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $n, k$ and $a$ be positive integers. The Stirling numbers of the first kind, denoted by $s(n,k)$, count the number of permutations of $n$ elements with $k$ disjoint cycles. Let $p$ be a prime. In recent years, Lengyel, Komatsu and Young, Leonetti and Sanna, Adelberg, Hong and Qiu made some progress in the study of the $p$-adic valuations of $s(n,k)$. In this paper, by using Washington's congruence on the generalized harmonic number and the $n$-th Bernoulli number $B_n$ and the properties of $m$-th Stirling numbers of the first kind obtained recently by the authors, we arrive at an exact expression or a lower bound of $v_p(s(ap, k))$ with $a$ and $k$ being integers such that $1\\le a\\le p-1$ and $1\\le k\\le ap$. This infers that for any regular prime $p\\ge 7$ and for arbitrary integers $a$ and $k$ with $5\\le a\\le p-1$ and $\\max\\{1,a-2\\}\\le k\\le ap-1$, one has $v_p(H(ap-1,k)) < -\\frac{\\log{(ap-1)}}{2\\log p}$ with $H(ap-1, k)$ being the $k$-th elementary symmetric function of $1, \\frac{1}{2}, ..., \\frac{1}{ap-1}$. This supports a conjecture of Leonetti and Sanna raised in 2017. We also present results on $v_p(s(ap^n,ap^n-k))$ from which one can derive that under certain condition, for any prime $p\\ge 5$, any odd number $k\\ge 3$ and any sufficiently large integer $n$, if $(a,p)=1$, then $v_p(s(ap^{n+1},ap^{n+1}-k))=v_p(s(ap^n,ap^n-k))+2$. It confirms partially Lengyel's conjecture proposed in 2015.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-15T15:44:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "first kind",
      "stirling numbers",
      "adic properties",
      "th elementary symmetric function",
      "confirms partially lengyels conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}