{
  "id": "2003.05939",
  "version": "v1",
  "published": "2020-03-12T11:13:42.000Z",
  "updated": "2020-03-12T11:13:42.000Z",
  "title": "Some new results about a conjecture by Brian Alspach",
  "authors": [
    "Simone Costa",
    "Marco Antonio Pellegrini"
  ],
  "categories": [
    "math.CO",
    "math.GR"
  ],
  "abstract": "In this short paper we consider the following conjecture, proposed by Brian Alspach, concerning partial sums in finite cyclic groups: given a subset $A$ of $\\mathbb{Z}_n\\setminus \\{0\\}$ of size $k$ such that $\\sum_{z\\in A} z\\not= 0$, it is possible to find an ordering $(a_1,\\ldots,a_k)$ of the elements of $A$ such that the partial sums $s_i=\\sum_{j=1}^i a_j$ are nonzero and $s_i\\neq s_j$ for all $1 \\leq i < j \\leq k$. This conjecture is known to be true for subsets of size $k\\leq 11$ in cyclic groups of prime order. We extend such result to any torsion-free abelian group and, as a consequence, we provide an asymptotic result in $\\mathbb{Z}_n$. Finally, we illustrate some connections with other related conjectures.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-12T11:13:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C25",
      "20K15"
    ],
    "keywords": [
      "brian alspach",
      "finite cyclic groups",
      "torsion-free abelian group",
      "prime order",
      "concerning partial sums"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}