{
  "abstract": "Let $M(m,n)$ and $N(m,n)$ denote the numbers of partitions of $n$ with crank $m$ and rank $m$, respectively. Ji and Zang proved that the crank sequence $(M(m,n))_{|m|\\le n-1}$ is unimodal for every $n\\ge44$. Bringmann, Man, and Rolen subsequently established the unimodality of the rank sequence $(N(m,n))_{|m|\\le n-2}$ for every $n\\ge39$. Ji and Zang also conjectured that $(M(m,n))_{|m|\\le n-71}$ is log-concave for every $n\\ge72$, and that $(N(m,n))_{|m|\\le n-72}$ is log-concave for every $n\\ge73$. In this paper, we prove both conjectures.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.06757",
  "author_total": 3,
  "authors": [
    {
      "name": "Jianxi Mao"
    },
    {
      "name": "Wenle Shi"
    },
    {
      "name": "Bao-Xuan Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d1f24e46mbgpk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.06757",
  "fulltext_url": null,
  "id": "d1f24e46mbgpk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.06757",
  "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"
    ],
    [
      "arxiv",
      "math.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.06757",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d1f24e46mbgpk.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.06757",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.06757"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Proof of two conjectures on the log-concavity of partition ranks and cranks",
  "title_text": "Proof of two conjectures on the log-concavity of partition ranks and cranks",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:14Z",
  "url": "https://research.draconitic.com/paper/d1f24e46mbgpk",
  "venue": "arXiv",
  "versions": []
}