{
  "abstract": "We investigate Conjecture 7.4 of Guo and Krattenthaler \\cite{GK}, which predicts that for every positive integer $n$, the polynomial \\[ P_n(q) = \\frac{(1-q)^2}{(1-q^{10n-1})(1-q^{15n-1})}\\qbin{30n}{5n} \\] has non-negative coefficients, except for $q^1$ and $q^{125n^2-25n+3}$, which equal $-1$. We study this problem through three main perspectives: First, by linking the boundary coefficients to the second differences of the partition function, we settle the first and last $25n+1$ coefficients via a novel partition injection. Second, by introducing a ``defect'' generalization of Andrews' classical lemma, we reduce the conjecture to an almost-unimodality property and employ a uniform saddle-point analysis to prove it analytically for all large $n \\ge 2\\cdot 10^{29}$. Finally, we connect these quotients to graph binomial coefficients, providing a natural orbit-counting interpretation via Burnside's lemma on directed cycles and paths.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.19249",
  "author_total": 2,
  "authors": [
    {
      "name": "Madjid Mirzavaziri"
    },
    {
      "name": "Daniel Yaqubi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/frg708s0bg7rx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.19249",
  "fulltext_url": null,
  "id": "frg708s0bg7rx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.19249",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7221994020",
  "pdf_url": "https://arxiv.org/pdf/2610.19249",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/frg708s0bg7rx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.19249",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.19249"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Graph binomial coefficients and a positivity conjecture of Guo and Krattenthaler",
  "title_text": "Graph binomial coefficients and a positivity conjecture of Guo and Krattenthaler",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/frg708s0bg7rx",
  "venue": "arXiv",
  "versions": []
}