{
  "id": "1301.7651",
  "version": "v3",
  "published": "2013-01-31T15:42:34.000Z",
  "updated": "2013-10-15T01:04:19.000Z",
  "title": "Some divisibility properties of binomial and q-binomial coefficients",
  "authors": [
    "Victor J. W. Guo",
    "C. Krattenthaler"
  ],
  "comment": "16 pages, add a note that Conjectures 7.2 and 7.3 are proved, to appear in J. Number Theory",
  "doi": "10.1016/j.jnt.2013.08.012",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "We first prove that if $a$ has a prime factor not dividing $b$ then there are infinitely many positive integers $n$ such that $\\binom {an+bn} {an}$ is not divisible by $bn+1$. This confirms a recent conjecture of Z.-W. Sun. Moreover, we provide some new divisibility properties of binomial coefficients: for example, we prove that $\\binom {12n} {3n}$ and $\\binom {12n} {4n}$ are divisible by $6n-1$, and that $\\binom {330n} {88n}$ is divisible by $66n-1$, for all positive integers $n$. As we show, the latter results are in fact consequences of divisibility and positivity results for quotients of $q$-binomial coefficients by $q$-integers, generalizing the positivity of $q$-Catalan numbers. We also put forward several related conjectures.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-10-15T01:04:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B65",
      "05A10",
      "05A30"
    ],
    "keywords": [
      "divisibility properties",
      "q-binomial coefficients",
      "positive integers",
      "conjecture",
      "catalan numbers"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1301.7651G"
    }
  }
}