{
  "abstract": "Let $a(n,k)$ be the number of $1324$-avoiding permutations of length $n$ with $k$ inversions. Linusson and Verkama proved $a(n,k)\\le a(n+1,k)$ for $k\\le2n-7$. We study the obstruction beyond that line: the residuals $\\mathcal R_{δ,n}$, namely the indecomposable, non-almost-decomposable avoiders at defect $δ=k-2n+7$. Contracting maximal increasing consecutive runs reduces residuality to a quadratic equation on a finite family of skeletons. It follows that, for every fixed $δ$, the eventual count has the form $|\\mathcal R_{δ,n}|=A_δn^2+B_δn+C_δ$. Our central uniform result determines the quadratic coefficient at every defect: with $P(q)=\\prod_{j\\ge1}(1-q^j)^{-1}$, $\\sum_{δ\\ge0}A_δq^δ=4q^3(1+q)P(q)^2/(1-q)^2$. This is a formula for the leading coefficient of the residual count, not for the full count. The same structural estimates give computer-assisted proofs of $a(n,k)\\le a(n+1,k)$ for every $n\\ge1$ and $k\\le2n+6$, and of regions whose width grows with $n$: for $n\\ge2^{16},2^{18},2^{20}$ the defect may be as large as $\\lfloor\\sqrt n/4\\rfloor$, $\\lfloor\\sqrt n/3\\rfloor$, $\\lfloor\\sqrt n/2\\rfloor$, respectively. More generally, every fixed $c<\\sqrt2\\log(5)/\\log(68)$ is admissible for all sufficiently large $n$. The three added fixed defects $11,12,13$ use complete catalogue and rational-sum certificates supplied in the accompanying archival supplement. The leading-coefficient theorem is obtained from a complete finite classification of marked rank-three cores and all-parameter extension lemmas. We also determine the exact rank-three stabilization onset, while keeping it separate from the still unknown onset of the complete residual count. The unrestricted Claesson--Jel'inek--Steingr'imsson conjecture, the full residual polynomials, and the sharp global base-length bound remain open.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06717",
  "author_total": 1,
  "authors": [
    {
      "name": "Lingsen Meng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bz8kaec2sjrgt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06717",
  "fulltext_url": null,
  "id": "bz8kaec2sjrgt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06717",
  "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": "W7212050003",
  "pdf_url": "https://arxiv.org/pdf/2609.06717",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bz8kaec2sjrgt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.06717",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06717"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Residual structure and growing inversion-monotonicity regions for 1324-avoiding permutations",
  "title_text": "Residual structure and growing inversion-monotonicity regions for 1324-avoiding permutations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/bz8kaec2sjrgt",
  "venue": "arXiv",
  "versions": []
}