{
  "id": "2506.14312",
  "version": "v2",
  "published": "2025-06-17T08:42:49.000Z",
  "updated": "2025-06-20T08:51:09.000Z",
  "title": "Schreier Sets of Multiples of an Integer, Linear Recurrence, and Pascal Triangle",
  "authors": [
    "Hung Viet Chu",
    "Zachary Louis Vasseur"
  ],
  "comment": "14 pages, 4 tables",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "A finite nonempty set $F$ is said to be Schreier (maximal Schreier, respectively) if $\\min F\\ge |F|$ ($\\min F = |F|$, respectively). For $k,n\\in\\mathbb{N}$, let $$s_{k,n}\\ :=\\ |\\{F\\subset\\{k, 2k,\\ldots, nk\\}\\,:\\, F\\mbox{ is Schreier and }nk\\in F\\}|.$$ We show that $(s_{k,n})_{n=1}^\\infty$ is a subsequence with terms taken periodically from the Padovan-like sequence $(a_{k,n})_{n=0}^\\infty$ defined as: $a_{k,0} = a_{k,1} = 1, a_{k, 2} = \\cdots = a_{k, k} = 2$, and $$a_{k, n}\\ =\\ a_{k,n-k} + a_{k,n-k-1},\\mbox{ for } n\\ge k+1.$$ As an application, we obtain an alternative proof of the linear recurrence of $(s_{k,n})_{n=1}^\\infty$ discovered by Beanland et al. Furthermore, a similar result holds for the sequence $(s^{(m)}_{k,n})_{n=1}^\\infty$ that counts maximal Schreier sets. Finally, we prove that $$s^{(m)}_{k,n}\\ =\\ 2s_{k,n}-s_{k,n+1}, \\mbox{ for all }n\\ge 1.$$",
  "revisions": [
    {
      "version": "v2",
      "updated": "2025-06-20T08:51:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A19",
      "11B37",
      "11Y55",
      "05A15"
    ],
    "keywords": [
      "linear recurrence",
      "pascal triangle",
      "counts maximal schreier sets",
      "similar result holds",
      "finite nonempty set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}