{
  "abstract": "Let $$ S_n=\\left\\langle \\genfrac{[}{]}{0pt}{}{n}{1},\\ldots,\\genfrac{[}{]}{0pt}{}{n}{n-1}\\right\\rangle $$ be the numerical semigroup generated by the nontrivial unsigned Stirling numbers of the first kind in the $n$th row. Writing $a=\\genfrac{[}{]}{0pt}{}{n}{n-1}=\\binom n2$ and $b_j=\\genfrac{[}{]}{0pt}{}{n}{n-2j}$, we determine the complete gcd filtration of the reduced generating system. More precisely, if \\[ M_j=\\frac12\\operatorname{lcm}\\{m\\ge1:φ(m)\\le2j\\}, \\] then $$ \\gcd(a,b_1,\\ldots,b_j)=\\frac{a}{\\gcd(a,M_j)}. $$ The proof uses prime-power block polynomials and exact $p$-adic support at threshold degrees. We then establish a divisor-support property for canonical mixed-radix reductions and combine it with an elementary-symmetric growth estimate to prove that, after inactive generators are removed, the resulting generating sequence is telescopic. Consequently, the Apéry set of $S_n$ is rectangular, yielding an explicit Frobenius formula. The same structure shows that $S_n$ is free and symmetric and gives explicit formulas for its genus and conductor, while its type is one.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03070",
  "author_total": 2,
  "authors": [
    {
      "name": "Takao Komatsu"
    },
    {
      "name": "Kyunghwan Song"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9r36r2w1q21rx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03070",
  "fulltext_url": null,
  "id": "9r36r2w1q21rx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03070",
  "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.NT"
    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7220372298",
  "pdf_url": "https://arxiv.org/pdf/2610.03070",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9r36r2w1q21rx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.03070",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03070"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Telescopic Structure of Numerical Semigroups Generated by Unsigned Stirling Numbers of the First Kind",
  "title_text": "Telescopic Structure of Numerical Semigroups Generated by Unsigned Stirling Numbers of the First Kind",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/9r36r2w1q21rx",
  "venue": "arXiv",
  "versions": []
}