{
  "id": "1402.3970",
  "version": "v1",
  "published": "2014-02-17T11:36:19.000Z",
  "updated": "2014-02-17T11:36:19.000Z",
  "title": "On Additive Combinatorics of Permutations of \\mathbb{Z}_n",
  "authors": [
    "L. Sunil Chandran",
    "Deepak Rajendraprasad",
    "Nitin Singh"
  ],
  "comment": "9 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $\\mathbb{Z}_n$ denote the ring of integers modulo $n$. In this paper we consider two extremal problems on permutations of $\\mathbb{Z}_n$, namely, the maximum size of a collection of permutations such that the sum of any two distinct permutations in the collection is again a permutation, and the maximum size of a collection of permutations such that the sum of any two distinct permutations in the collection is not a permutation. Let the sizes be denoted by $s(n)$ and $t(n)$ respectively. The case when $n$ is even is trivial in both the cases, with $s(n)=1$ and $t(n)=n!$. For $n$ odd, we prove $s(n)\\geq (n\\phi(n))/2^k$ where $k$ is the number of distinct prime divisors of $n$. When $n$ is an odd prime we prove $s(n)\\leq \\frac{e^2}{\\pi} n ((n-1)/e)^\\frac{n-1}{2}$. For the second problem, we prove $2^{(n-1)/2}.(\\frac{n-1}{2})!\\leq t(n)\\leq 2^k.(n-1)!/\\phi(n)$ when $n$ is odd.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-02-17T11:36:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A05",
      "11A07",
      "11A41",
      "05E99",
      "05D05"
    ],
    "keywords": [
      "additive combinatorics",
      "collection",
      "distinct permutations",
      "distinct prime divisors",
      "extremal problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.3970C"
    }
  }
}