{
  "abstract": "We study the distribution of the number of double deficiencies (DD) in permutations of length n avoiding one or two patterns of length 3. Using structural decompositions of these avoidance classes--together with a lattice-path decomposition in the 321-avoiding case--we derive functional equations and convolution-type recurrences that efficiently compute the corresponding double-deficiency generating functions in all but one single-pattern case. In the 321-avoiding permutations, the resulting generating function is algebraic; we derive exact formulas for the mean and variance and prove that the distribution is close in total variation to Bin(n-2,1/4), with an explicit convergence rate. We also identify a DD-preserving symmetry that yields DD-Wilf equivalences, reducing the number of two-pattern cases that need to be considered separately. For the resulting two-pattern classes, we obtain explicit recurrences, including C-finite relations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10492",
  "author_total": 3,
  "authors": [
    {
      "name": "Tipaluck Krityakierne"
    },
    {
      "name": "Thotsaporn Aek Thanatipanonda"
    },
    {
      "name": "Doron Zeilberger"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9900s5wn325qw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10492",
  "fulltext_url": null,
  "id": "9900s5wn325qw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10492",
  "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": "W7212157628",
  "pdf_url": "https://arxiv.org/pdf/2609.10492",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9900s5wn325qw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:16Z",
      "native_id": "2609.10492",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10492"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The Distribution of Double Deficiencies in Pattern-Avoiding Permutations",
  "title_text": "The Distribution of Double Deficiencies in Pattern-Avoiding Permutations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:16Z",
  "url": "https://research.draconitic.com/paper/9900s5wn325qw",
  "venue": "arXiv",
  "versions": []
}