{
  "id": "2203.16658",
  "version": "v1",
  "published": "2022-03-30T20:23:31.000Z",
  "updated": "2022-03-30T20:23:31.000Z",
  "title": "On Sequences in Cyclic Groups with Distinct Partial Sums",
  "authors": [
    "Simone Costa",
    "Stefano Della Fiore",
    "M. A. Ollis",
    "Sarah Z. Rovner-Frydman"
  ],
  "comment": "19 pages, plus supporting tables and code",
  "categories": [
    "math.CO"
  ],
  "abstract": "A subset of an abelian group is {\\em sequenceable} if there is an ordering $(x_1, \\ldots, x_k)$ of its elements such that the partial sums $(y_0, y_1, \\ldots, y_k)$, given by $y_0 = 0$ and $y_i = \\sum_{j=1}^i x_i$ for $1 \\leq i \\leq k$, are distinct, with the possible exception that we may have $y_k = y_0 = 0$. We demonstrate the sequenceability of subsets of size $k$ of $\\mathbb{Z}_n \\setminus \\{ 0 \\}$ when $n = mt$ in many cases, including when $m$ is either prime or has all prime factors larger than $k! /2$ for $k \\leq 11$ and $t \\leq 5$ and for $k=12$ and $t \\leq 4$. We obtain similar, but partial, results for $13 \\leq k \\leq 15$. This represents progress on a variety of questions and conjectures in the literature concerning the sequenceability of subsets of abelian groups, which we combine and summarize into the conjecture that if a subset of an abelian group does not contain 0 then it is sequenceable.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-30T20:23:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C25"
    ],
    "keywords": [
      "distinct partial sums",
      "cyclic groups",
      "abelian group",
      "prime factors larger",
      "sequenceability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}