{
  "abstract": "Let a_A(n) be the number of permutations of {1,...,n} all of whose cycle lengths lie in a set A. We develop a valuation calculus for these numbers. Mobius inversion writes the exponential generating function as an Euler product, and at a prime p the factors with non-p-integral exponents, the blocks, determine a finite knapsack problem. Its optimal value gives a lower bound for v_p(a_A(n)), and the bound is attained exactly when one residue modulo p, the sum of the terms of the optimal choices, does not vanish. For a fixed set A and prime p the bounds and residues for all n <= N cost O(N^2) operations. When one block is strictly the most efficient, the valuation inside each residue class is that of a locally analytic p-adic function. We use the calculus to determine the 2-adic and 3-adic valuations for cycle lengths at most 3, 4 and 5, to prove the divisibility conjectures recorded in A378495, A079128, A193437 and A193438 and the 3-adic statements recorded in A057693 and A070945, and to treat 99 OEIS sequences in all. We also prove sharp divisibility and gcd theorems, show that at n = 111 the gcd of the counts over all sets of excluded cycle lengths containing {1,2,3} is one third of the gcd over the sets {1,2,3,k}, and determine eventual divisibility and tail gcds modulo composite integers. Part of the paper is formalized in Lean 4.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.40342",
  "author_total": 1,
  "authors": [
    {
      "name": "Ahaan Kallat"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6pxhqr2xfcew5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.40342",
  "fulltext_url": null,
  "id": "6pxhqr2xfcew5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.40342",
  "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"
    ],
    [
      "arxiv",
      "math.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.40342",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6pxhqr2xfcew5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.40342",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.40342"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A p-adic Valuation Calculus for Permutations with Restricted Cycle Lengths",
  "title_text": "A p-adic Valuation Calculus for Permutations with Restricted Cycle Lengths",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/6pxhqr2xfcew5",
  "venue": "arXiv",
  "versions": []
}