{
  "abstract": "Let $\\mathcal{P}_n$ denote the set of distinct permutations of length $n$ that arise from the flattening process applied to the partitions of $[n]=\\{1,\\ldots,n\\}$. In this paper, we consider the problem of avoidance of a single classical pattern of length four by members of $\\mathcal{P}_n$. Let $p_n(τ)$ denote the number of members of $\\mathcal{P}_n$ that avoid the pattern $τ$. We show that $p_n(τ)=C_{n-1}$ for all $n \\geq 1$ for seven patterns of length four yielding new combinatorial interpretations of the Catalan number sequence. Further, we show that $p_n(τ)$ corresponds to the binomial transform of Catalan numbers for three other patterns. To establish our results, we suitably refine the counting sequence $p_n(τ)$ in each case so as to obtain a system of functional equations satisfied by the corresponding generating functions. These functional equations may then be solved explicitly leading to a determination of $p_n(τ)$ in each case.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.11271",
  "author_total": 3,
  "authors": [
    {
      "name": "Toufik Mansour"
    },
    {
      "name": "Olivia Nabawanda"
    },
    {
      "name": "Mark Shattuck"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dqkjdrza8egnq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.11271",
  "fulltext_url": null,
  "id": "dqkjdrza8egnq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.11271",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7221133303",
  "pdf_url": "https://arxiv.org/pdf/2610.11271",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dqkjdrza8egnq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.11271",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.11271"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Distinct flattened partitions avoiding a pattern of length four",
  "title_text": "Distinct flattened partitions avoiding a pattern of length four",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/dqkjdrza8egnq",
  "venue": "arXiv",
  "versions": []
}