{
  "abstract": "For positive integers $a\\le b\\le c\\le d$ with $ad=bc$, we prove that the Lucas binomial coefficients satisfy $(-1)^{a+1}(L_{b,c}[X]-L_{a,d}[X])\\in\\mathbb{N}[e_1[X],e_2[X],h_2[X]]$, resolving Bergeron's conjecture on Schur positivity in two variables. We also prove the same positivity for $L_{u,v}[X]-e_1[X]^{uv-rs}L_{r,s}[X]$ when $rs<uv$. Using Lucas atoms, we establish $e$-positivity for degree-adjusted Lucas multinomial differences under dominance order.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.35465",
  "author_total": 4,
  "authors": [
    {
      "name": "Lilan Dai"
    },
    {
      "name": "Dun Qiu"
    },
    {
      "name": "Weichen Wang"
    },
    {
      "name": "Minhao Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1e68055w07559.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.35465",
  "fulltext_url": null,
  "id": "1e68055w07559",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.35465",
  "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",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7223856961",
  "pdf_url": "https://arxiv.org/pdf/2610.35465",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1e68055w07559.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.35465",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.35465"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The Schur positivity of Lucas binomial differences",
  "title_text": "The Schur positivity of Lucas binomial differences",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/1e68055w07559",
  "venue": "arXiv",
  "versions": []
}