{
  "abstract": "We study permutations $π\\in S_n$ that avoid $123$ while satisfying the adjacency constraint $\\lvert π_{i+1}-π_i\\rvert \\le m$ for some $m\\in\\mathbb{Z}^{+}$. We first show that every admissible permutation is localized within vertical distance $m$ of the reverse diagonal, revealing a strong global restriction imposed by the interaction between pattern avoidance and bounded adjacency. For every fixed $m$, we then construct an exact finite-state description of the family, with precisely $2^{m+1}-m-1$ realizable states. This yields an explicit transfer-matrix enumeration and, in particular, a rational generating function for every fixed $m$; consequently, the enumeration sequence satisfies an eventual constant-coefficient linear recurrence. We further identify the exponential growth rate with the spectral radius $ρ_m$ of the corresponding transfer matrix and show that $ρ_m<4$, is nondecreasing in $m$, and converges to the Catalan growth constant $4$ as $m\\to\\infty$. Finally, we show that the bounded-adjacency constraint breaks the ordinary length-three Wilf equivalence, and we develop algorithmic and graph-theoretic interpretations of the family, including a correspondence with constrained Hamiltonian paths in powers of the path graph.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.02526",
  "author_total": 1,
  "authors": [
    {
      "name": "Nathaniel Nadler"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/azcatgpwxmxn3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02526",
  "fulltext_url": null,
  "id": "azcatgpwxmxn3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.02526",
  "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": "W7220387094",
  "pdf_url": "https://arxiv.org/pdf/2610.02526",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/azcatgpwxmxn3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.02526",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02526"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "123-Avoiding permutations with an adjacency constraint",
  "title_text": "123-Avoiding permutations with an adjacency constraint",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/azcatgpwxmxn3",
  "venue": "arXiv",
  "versions": []
}