{
  "id": "2301.09367",
  "version": "v1",
  "published": "2023-01-23T11:21:58.000Z",
  "updated": "2023-01-23T11:21:58.000Z",
  "title": "Sequencings in Semidirect Products via the Polynomial Method",
  "authors": [
    "Simone Costa",
    "Stefano Della Fiore",
    "M. A. Ollis"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:2203.16658",
  "categories": [
    "math.CO"
  ],
  "abstract": "The partial sums of a sequence ${\\mathbf x} = x_1, x_2, \\ldots, x_k$ of distinct non-identity elements of a group $(G,\\cdot)$ are $s_0 = id_G$ and $s_j = \\prod_{i=1}^j x_i$ for $0 < j \\leq k$. If the partial sums are all different then ${\\mathbf x}$ is a linear sequencing and if the partial sums are all different when $|i-j| \\leq t$ then ${\\mathbf x}$ is a $t$-weak sequencing. We investigate these notions of sequenceability in semidirect products using the polynomial method. We show that every subset of order $k$ of the non-identity elements of the dihedral group of order $2m$ has a linear sequencing when $k \\leq 12$ and either $m>3$ is prime or every prime factor of $m$ is larger than $k!$, unless $s_k$ is unavoidably the identity; that every subset of order $k$ of a non-abelian group of order three times a prime has a linear sequencing when $5 < k \\leq 10$, unless $s_k$ is unavoidably the identity; and that if the order of a group is $pe$ then all sufficiently large subsets of the non-identity elements are $t$-weakly sequenceable when $p>3$ is prime, $e \\leq 3$ and $t \\leq 6$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-23T11:21:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "polynomial method",
      "semidirect products",
      "partial sums",
      "linear sequencing",
      "distinct non-identity elements"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}