{
  "abstract": "Let $\\mathcal I_n^B$ be the involutions of the hyperoctahedral group $\\mathfrak B_n$, and let $\\des^B$ denote the descent number with respect to the natural Coxeter order. We derive the fixed-point-refined Worpitzky identity \\[ \\sum_{n\\ge0}\\frac{\\mathcal F_n(p,t)\\,z^n}{(1-t)^{n+1}} =\\sum_{m\\ge0} \\frac{(1+pz)^m\\,t^m}{(1-pz)^{m+1}(1-z^2)^{m(m+1)}}, \\quad \\mathcal F_n(p,t)=\\sum_{π\\in\\mathcal I_n^B}p^{\\fixB(π)}t^{\\des^B(π)}. \\] Extracting the stratum with $j$ two-cycles and $f$ fixed positions yields a one-parameter deformation of the fixed-point-free Worpitzky series of Wan, Gao, Li and Yang. After the change of variables $x=t/(1+t)^2$, this deformation becomes a positive binomial transform. More precisely, if \\[ P_j(x)=\\sum_sD_{2j,s}x^s \\] is the fixed-point-free $γ$-polynomial, then the transform coefficients $A_{j,r}(x)$ are determined by \\[ \\sum_{r\\ge0}A_{j,r}(x)W^r =\\sum_{s=0}^{j}D_{2j,s}x^s (1+4xW)^{2j-2s}(1+2W+4xW^2)^s, \\] and the $γ$-polynomial of the $(j,f)$-stratum is \\[ Φ_{j,f}(x)=\\sum_{r=0}^{f}\\binom fr A_{j,r}(x). \\] This manifestly positive transform is the main structural result of the paper. As consequences, every fixed cycle-type stratum is $γ$-positive and $\\mathcal F_n(p,t)$ is coefficientwise $γ$-positive in the fixed-point variable $p$. The cases $f=0$ and $p=1$ recover, respectively, the fixed-point-free theorem of Wan--Gao--Li--Yang and the all-involution theorem of Cao--Liu. We also give explicit formulas for the first binomial layers and for the strata with one and two two-cycles.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.04922",
  "author_total": 1,
  "authors": [
    {
      "name": "Jiang Zeng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ah0w2k9h2q9g8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.04922",
  "fulltext_url": null,
  "id": "ah0w2k9h2q9g8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.04922",
  "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/2609.04922",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ah0w2k9h2q9g8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:18Z",
      "native_id": "2609.04922",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.04922"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A Fixed-Point Worpitzky Identity and a Positive Binomial Transform for Type $B$ Involutions",
  "title_text": "A Fixed-Point Worpitzky Identity and a Positive Binomial Transform for Type B Involutions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:18Z",
  "url": "https://research.draconitic.com/paper/ah0w2k9h2q9g8",
  "venue": "arXiv",
  "versions": []
}