{
  "abstract": "Let $\\cR=\\{r+jm:j\\geq0\\}$ and $\\cS=\\{s+jm:j\\geq0\\}$ be disjoint arithmetic progressions of positive integers, and choose a partition of $n$ uniformly at random. We determine the limiting probability that the number of parts in $\\cR$ is larger than the number of parts in $\\cS$. More precisely, if $a=r/m$ and $b=s/m$, then this probability converges to $I_{1/2}(a,b)$, the regularized incomplete beta function. We prove the stronger distributional statement that, after multiplication by the partition saddle-point parameter, the difference of the two numbers of parts converges to $m^{-1}(\\log G_b-\\log G_a)$, where $G_a$ and $G_b$ are independent gamma variables. The proof uses the Boltzmann conditioning device for random partitions, a $q$-gamma evaluation of the unconditioned characteristic function, and a uniform Fourier estimate showing that conditioning on the total size does not change the limit. We also evaluate the finite-truncation limit studied by Li and Zhao. The two evaluations coincide, proving their conjecture on the commutation of the truncation and large-size limits.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.30700",
  "author_total": 2,
  "authors": [
    {
      "name": "Jiyou Li"
    },
    {
      "name": "Sicheng Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cjnde7ckd2dzy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.30700",
  "fulltext_url": null,
  "id": "cjnde7ckd2dzy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.30700",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7223290000",
  "pdf_url": "https://arxiv.org/pdf/2610.30700",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cjnde7ckd2dzy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.30700",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.30700"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "On Arithmetic-Progression Biases in Random Partitions: Limiting Laws and the Li-Zhao Conjecture",
  "title_text": "On Arithmetic-Progression Biases in Random Partitions: Limiting Laws and the Li-Zhao Conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/cjnde7ckd2dzy",
  "venue": "arXiv",
  "versions": []
}