{
  "id": "2003.02511",
  "version": "v1",
  "published": "2020-03-05T10:01:19.000Z",
  "updated": "2020-03-05T10:01:19.000Z",
  "title": "On zero-sum free sequences contained in random subsets of finite cyclic groups",
  "authors": [
    "Sang June Lee",
    "Jun Seok Oh"
  ],
  "comment": "15 pages",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "Let $C_n$ be a cyclic group of order $n$. A sequence $S$ of length $\\ell$ over $C_n$ is a sequence $S = a_1\\boldsymbol\\cdot a_2\\boldsymbol\\cdot \\ldots\\boldsymbol\\cdot a_{\\ell}$ of $\\ell$ elements in $C_n$, where a repetition of elements is allowed and their order is disregarded. We say that $S$ is a zero-sum sequence if $\\Sigma_{i=1}^{\\ell} a_i = 0$ and that $S$ is a zero-sum free sequence if $S$ contains no zero-sum subsequence. Let $R$ be a random subset of $C_n$ obtained by choosing each element in $C_n$ independently with probability $p$. Let $N^R_{n-1-k}$ be the number of zero-sum free sequences of length $n-1-k$ in $R$. Also, let $N^R_{n-1-k,d}$ be the number of zero-sum free sequences of length $n-1-k$ having $d$ distinct elements in $R$. We obtain the expectation of $N^R_{n-1-k}$ and $N^R_{n-1-k,d}$ for $0\\leq k\\leq \\big\\lfloor \\frac{n}{3} \\big\\rfloor$. We also show a concentration result on $N^R_{n-1-k}$ and $N^R_{n-1-k,d}$ when $k$ is fixed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-05T10:01:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B50",
      "11B30",
      "05D40"
    ],
    "keywords": [
      "zero-sum free sequence",
      "finite cyclic groups",
      "random subset",
      "distinct elements",
      "zero-sum subsequence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}