{
  "abstract": "The basic hypergeometric series ${}_{n+1}ϕ_{n}$ whose upper and lower parameters are non-negative integer powers of $q$ satisfying a certain constraint can be, after a normalisation, computed as a positive polynomial. We derive three formulas for this polynomial. The first formula is a generalisation of the MacMahon identity in which an infinite binomial series is identified with a normalised generating function of des and maj statistics of multiset permutations. The second formula is a generating function of multiset permutations for the inversion statistic and a generalisation of the statistic mstc. The third formula is a finite sum containing $2n$ $q$-binomial coefficients.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07549",
  "author_total": 1,
  "authors": [
    {
      "name": "Alexandr Garbali"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3nh0dtkv9ye98.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07549",
  "fulltext_url": null,
  "id": "3nh0dtkv9ye98",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07549",
  "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-ph"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7220940573",
  "pdf_url": "https://arxiv.org/pdf/2610.07549",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3nh0dtkv9ye98.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:22Z",
      "native_id": "2610.07549",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07549"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Positivity of the basic hypergeometric series and the MacMahon identity",
  "title_text": "Positivity of the basic hypergeometric series and the MacMahon identity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/3nh0dtkv9ye98",
  "venue": "arXiv",
  "versions": []
}