{
  "abstract": "For a nonnegative integer $k$ and a rational number $r\\in\\mathbb{Q}^+$, we define the generalized Gaussian binomial coefficient $\\qbinom{r+k}{k} = \\frac{(q^{r+1}; q)_k}{(q; q)_k}$. When $r=a/b$ with $a,b$ coprime positive integers and $b\\geq 2$, expanding $\\qbinom{r+k}{k}$ via the finite $q$-binomial theorem produces fractional powers of $q$, so that $\\qbinom{r+k}{k}$ is a \\emph{Puiseux series} in $q$ with nonnegative exponents; concretely it lies in $\\mathbb{Q}[[q^{1/b}]]$. The notion we single out is the \\emph{integer trace} of this expansion, the subseries consisting of those terms $c_r(d)\\,q^d$ whose exponent $d$ is an integer, with all fractional powers discarded. This projection is not standard, and there is no a~priori reason for the surviving coefficients to behave coherently as $r$ varies. Nonetheless, ordering the family by the coefficientwise partial order leads to the \\emph{$\\tfrac{1}{2}$-Conjecture}: among all $r\\in\\mathbb{Q}^+$, the value $r=\\tfrac{1}{2}$ maximizes the integer trace, in the sense that the coefficients of $\\qbinom{1/2+k}{k}$ dominate those of $\\qbinom{r+k}{k}$ coefficientwise for every $r$. That so elementary a definition should single out $\\tfrac{1}{2}$ this cleanly came as a surprise to us. We prove the conjecture in several special cases and provide further computational evidence.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.01919",
  "author_total": 2,
  "authors": [
    {
      "name": "Guo-Niu Han"
    },
    {
      "name": "Huan Xiong"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8544pp2v4sfyh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.01919",
  "fulltext_url": null,
  "id": "8544pp2v4sfyh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.01919",
  "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"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7163202108",
  "pdf_url": "https://arxiv.org/pdf/2606.01919",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8544pp2v4sfyh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2606.01919",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.01919"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The 1/2-Conjecture for $q$-Binomial Coefficients with Fractional Index",
  "title_text": "The 1/2-Conjecture for q-Binomial Coefficients with Fractional Index",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/8544pp2v4sfyh",
  "venue": "arXiv",
  "versions": []
}