{
  "abstract": "Let p(n,k) denote the number of partitions of n into exactly k parts. In 1942 Auluck, Chowla and Gupta conjectured that for every n the sequence p(n,1), p(n,2), ..., p(n,n) is unimodal. Szekeres proved this for all sufficiently large n, but with a threshold that was never made explicit. We prove the conjecture for every n >= 1. For n >= 10^5 we give an analytic argument with fully explicit constants: an exact probabilistic representation of the ratio p(n,k+1)/p(n,k), a second-order saddle-point main term with a rigorous uniform error bound in the transition window, and separate elementary or saddle-point estimates in the two tails. Every numerical constant in this argument is certified in ball arithmetic. For n <= 2*10^5 the conjecture is verified exhaustively by two independent programs, one of them using exact integer arithmetic only. Since log-concavity of k -> p(n,k) fails for every n >= 4, the argument works with first differences and never uses log-concavity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14084",
  "author_total": 1,
  "authors": [
    {
      "name": "Jaideep Sai Padhi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bwakvpnrw0t7q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14084",
  "fulltext_url": null,
  "id": "bwakvpnrw0t7q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14084",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7221812491",
  "pdf_url": "https://arxiv.org/pdf/2610.14084",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bwakvpnrw0t7q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.14084",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14084"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A proof of the Auluck-Chowla-Gupta Conjecture",
  "title_text": "A proof of the Auluck-Chowla-Gupta Conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/bwakvpnrw0t7q",
  "venue": "arXiv",
  "versions": []
}