{
  "id": "2205.14260",
  "version": "v1",
  "published": "2022-05-27T22:37:55.000Z",
  "updated": "2022-05-27T22:37:55.000Z",
  "title": "A Note on the Fibonacci Sequence and Schreier-type Sets",
  "authors": [
    "Hung Viet Chu"
  ],
  "comment": "5 pages, 0 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "A set $A$ of positive integers is said to be Schreier if either $A = \\emptyset$ or $\\min A\\ge |A|$. For $n\\ge 1$, let $$\\mathcal{K}_n\\ :=\\ \\{A\\subset \\{1, \\ldots, n\\}\\,:\\, \\mbox{either }A = \\emptyset \\mbox{ or } (\\max A-1\\in A\\mbox{ and }\\min A\\ge |A|)\\}.$$ We give a combinatorial proof that for each $n\\ge 1$, we have $|\\mathcal{K}_n| = F_n$, where $F_n$ is the $n$th Fibonacci number. As a corollary, we obtain a new combinatorial interpretation for the sequence $(F_n + n)_{n=1}^\\infty$. Next, we give a bijective proof for the recurrence of the sequence $(|\\mathcal{K}_{n, p, q}|)_{n=1}^\\infty$ (for fixed $p\\ge 1$ and $q\\ge 2$), where $\\mathcal{K}_{n, p, q} = $ $$\\{A\\subset \\{1, \\ldots, n\\}\\,:\\, \\mbox{either }A = \\emptyset \\mbox{ or } (\\max A-\\max_2 A = p\\mbox{ and }\\min A\\ge |A|\\ge q)\\}$$ and $\\max_2 A$ is the second largest integer in $A$, given that $|A|\\ge 2$. Note that by definition, $\\mathcal{K}_{n, 1, 2} = \\mathcal{K}_{n}$ for all $n\\ge 1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-27T22:37:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B39",
      "11B37",
      "11Y55"
    ],
    "keywords": [
      "fibonacci sequence",
      "schreier-type sets",
      "second largest integer",
      "th fibonacci number",
      "combinatorial interpretation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}