{
  "abstract": "A finite nonempty set $F\\subset\\mathbb{N}$ is Schreier if $\\min F\\ge |F|$. First, we prove a linear recurrence relation and compute initial counts for Schreier sets consisting of intervals. Two intervals of integers are separated if their union is not an interval. If $\\mathcal J_{k,n}$ is the collection of Schreier sets that are the union of exactly $k$ separated intervals, then the sequence $(|\\mathcal{J}_{k,n}|)_{n=1}^\\infty$ satisfies the characteristic polynomial $p_k(x) = (x-1)^{2k+1}(x+1)^k$. Furthermore, we introduce the new concept of $k$-super-Schreier sets and let $\\mathcal{S}_{k,n}$ denote the collection of $k$-super-Schreier sets whose maximum is $n$. We show that the sequence $(|\\mathcal{S}_{k,n}|)_{n=1}^\\infty$ satisfies a Fibonacci-type recurrence with a remainder term expressible as a polynomial of $n$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.06076",
  "author_total": 4,
  "authors": [
    {
      "name": "Hung Viet Chu"
    },
    {
      "name": "Mariam Khaduri"
    },
    {
      "name": "Moiz M. Khokhar"
    },
    {
      "name": "Ruoan Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cd4dpeqrktzn3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.06076",
  "fulltext_url": null,
  "id": "cd4dpeqrktzn3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.06076",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7201838342",
  "pdf_url": "https://arxiv.org/pdf/2608.06076",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cd4dpeqrktzn3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2608.06076",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.06076"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Schreier Sets of Intervals, Super-Schreier Sets, and Catalan Numbers",
  "title_text": "Schreier Sets of Intervals, Super-Schreier Sets, and Catalan Numbers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/cd4dpeqrktzn3",
  "venue": "arXiv",
  "versions": []
}