{
  "abstract": "We prove that every cluster monomial of type $A_n$ is log-concave and unimodal with respect to its Laurent expansion in a fan initial cluster. We derive a coefficient formula for laminar interval products arising from compatible diagonals in the polygon model. In even rank, we reduce log-concavity in fan coordinates to log-concavity along integer lines. In odd rank, we use convolution to handle the coefficient sums arising from the fan substitution. This gives partial affirmative answers to the log-concavity conjecture of Chen-Huang-Sun and the unimodality conjecture of Chen. Our result also complements Shen's study on the supports of products of canonical basis elements for cluster X-varieties of type $A_n$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.09312",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhichao Chen"
    },
    {
      "name": "Zhe Sun"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4xb7p4g0e3bkf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.09312",
  "fulltext_url": null,
  "id": "4xb7p4g0e3bkf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.09312",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
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  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7221046145",
  "pdf_url": "https://arxiv.org/pdf/2610.09312",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4xb7p4g0e3bkf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.09312",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.09312"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Log-concavity and unimodality of cluster monomials in fan coordinates of type $A_n$",
  "title_text": "Log-concavity and unimodality of cluster monomials in fan coordinates of type A_n",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:44Z",
  "url": "https://research.draconitic.com/paper/4xb7p4g0e3bkf",
  "venue": "arXiv",
  "versions": []
}