{
  "abstract": "For integers $r\\ge2$ and $n\\ge1$, let $$ F_{r,n}(q)=\\prod_{k=1}^{n}\\frac{1-q^{rk}}{1-q^k}. $$ The coefficient of $q^j$ in \\(F_{r,n}(q)\\) counts partitions of $j$ into parts at most $n$, each occurring at most $r-1$ times. Hughes proved that $F_{2,n}(q)=\\prod_{k=1}^{n}(1+q^k)$ is unimodal for every $n\\ge 1$. This result was reproved by Stanley using an algebraic approach and Odlyzko and Richmond using an analytic approach. Almkvist conjectured that $F_{r,n}(q)$ is unimodal in the following two cases: every even $r$ and every $n\\ge 1$; every odd $r$ and every $n\\ge 11$. He proved that this conjecture is true for $3\\le r \\le 20$ and $r=100,101$. In this paper, we completely settle the conjecture.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23779",
  "author_total": 3,
  "authors": [
    {
      "name": "Jianxi Mao"
    },
    {
      "name": "Wenle Shi"
    },
    {
      "name": "Bao-xuan Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2eyxmcp3154vd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23779",
  "fulltext_url": null,
  "id": "2eyxmcp3154vd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23779",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7214015452",
  "pdf_url": "https://arxiv.org/pdf/2609.23779",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2eyxmcp3154vd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2609.23779",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23779"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Proof of Almkvist's conjecture on the unimodality of partition polynomials",
  "title_text": "Proof of Almkvist's conjecture on the unimodality of partition polynomials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/2eyxmcp3154vd",
  "venue": "arXiv",
  "versions": []
}