{
  "abstract": "Let $P_n(q)=\\prod_{j=1}^n(1-q^{3j-2})(1-q^{3j-1})$. For each $p\\in\\{4,5,6,7,8\\}$, we prove that the coefficients of $P_n(q)^p$ in residue class $0$ modulo $3$ are nonnegative, and that those in residue class $2$, after zero terms are omitted, change sign exactly once, from positive to negative. The residue-$0$ coefficients are in fact strictly positive for $5\\le p\\le8$. This proves Conjecture 2.1 of Berkovich and Dhar in full. We further determine the limiting transition constants and a four-term asymptotic expansion for the transition centres. For $n\\ge301$, the signs in residue class $2$ are determined outside an interval of length $2\\varepsilon_p/n^2$ centred at $α_p n^2+β_p n+γ_p+δ_p/n$, where $\\varepsilon_4=1500$ and $\\varepsilon_p=150$ for $5\\le p\\le8$, and all constants admit explicit analytic definitions. The proof combines a corrected saddle relation, higher-order expansions resolving cancellation between the two dominant saddle contributions, and a rescaled positive-integral argument that uniformly controls the small-degree range.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.08496",
  "author_total": 1,
  "authors": [
    {
      "name": "Shutao Jiang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5xj00bzgz4nnn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.08496",
  "fulltext_url": null,
  "id": "5xj00bzgz4nnn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.08496",
  "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"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7220982657",
  "pdf_url": "https://arxiv.org/pdf/2610.08496",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5xj00bzgz4nnn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.08496",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.08496"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A proof of the Berkovich-Dhar conjecture modulo three",
  "title_text": "A proof of the Berkovich-Dhar conjecture modulo three",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/5xj00bzgz4nnn",
  "venue": "arXiv",
  "versions": []
}