{
  "abstract": "Gray, Payne, Swisher, and Watson conjectured that, for fixed integers $j \\geq 0$ and $k \\geq 2$, the number $FD_{j,k}(n)$ of partitions of perimeter $n$ having exactly $j$ part sizes of multiplicity at least $k$ is eventually at least the number $FO_{j,k}(n)$ having exactly $j$ distinct occurring part sizes divisible by $k$. We derive bivariate generating functions for both statistics using the profile-word encoding of a partition. For $k \\geq 3$, coefficient extraction shows that fixing $j$ changes the order of the dominant pole but not its location. The corresponding dominant singularities are positive real numbers $ρ_k$ and $σ_k$, where $ρ_k$ satisfies $ρ_k+ρ_k^2+\\cdots+ρ_k^k=1$ and $σ_k$ satisfies $σ_k^k=(1-σ_k)^{k-1}$. We prove that $ρ_k<σ_k$ for every $k \\geq 3$. Consequently, $FD_{j,k}(n)/FO_{j,k}(n) \\to \\infty$ as $n \\to \\infty$, proving the conjecture and yielding a strict eventual inequality for $k \\geq 3$. The case $k=2$ recovers the known exact identity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12005",
  "author_total": 1,
  "authors": [
    {
      "name": "Marcus McCrea"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9fmcz4phbz4b2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12005",
  "fulltext_url": null,
  "id": "9fmcz4phbz4b2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12005",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7212708499",
  "pdf_url": "https://arxiv.org/pdf/2609.12005",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9fmcz4phbz4b2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.12005",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12005"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Fixed-Perimeter Franklin Statistics and an Eventual Inequality",
  "title_text": "Fixed-Perimeter Franklin Statistics and an Eventual Inequality",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/9fmcz4phbz4b2",
  "venue": "arXiv",
  "versions": []
}