{
  "abstract": "We extend cyclotomic Euler--Mahonian factorization to colored permutation groups for two inversion statistics: Biagioli--Zeng length and the inversion number of the underlying unsigned permutation. At a primitive $m$th root and rank $n=mk+i$, each normalized enumerator is a Hadamard product of a rank-$i$ colored enumerator and a weighted rank-$k$ type-B enumerator. Separate color variables are retained. The type-B factors agree for the two statistics, yielding equidistribution modulo the rank jointly with descent number, major index and negative-entry number. We also determine their exact changes under positive fixed-point insertion and obtain rational kernels retaining descent and major index. These kernels mark arbitrary fixed-point numbers and give derangement expansions at every inversion weight. Signed specializations recover type-A and type-B identities and give colored flag-major refinements. Inverse statistics and color-sum subgroups admit related insertion formulas.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.05848",
  "author_total": 1,
  "authors": [
    {
      "name": "Jiang Zeng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2r291dm590rd9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.05848",
  "fulltext_url": null,
  "id": "2r291dm590rd9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.05848",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.05848",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2r291dm590rd9.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.05848",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.05848"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Cyclotomic factorization and fixed-point insertion in colored permutation groups",
  "title_text": "Cyclotomic factorization and fixed-point insertion in colored permutation groups",
  "type": "preprint",
  "updated_at": "2026-11-05T15:21:00Z",
  "url": "https://research.draconitic.com/paper/2r291dm590rd9",
  "venue": "arXiv",
  "versions": []
}