{
  "abstract": "We realize classical cumulants and their fixed-point equations in the double tensor unshuffle Hopf algebra introduced by Ebrahimi-Fard and Patras, placing classical, free, Boolean, and monotone cumulants in a common algebraic framework. A concatenation perturbation of the functionals yields left and right half-shuffle equations on ordinary words without changing the coproduct or either half-coproduct. Both equations are equivalent to the classical moment--cumulant formula and uniquely determine the classical cumulants. We also obtain a Hopf-algebraic realization of the classical-to-free cumulant transform through convolution exponentiation and the left half-shuffle logarithm. A commutative diagram identifies its partition-level action with summation over non-crossing closure fibers, recovering Lehner's closure-fiber formula within the framework of the double tensor unshuffle Hopf algebra.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33382",
  "author_total": 2,
  "authors": [
    {
      "name": "Xing Gao"
    },
    {
      "name": "Jingtao Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5kstzzfr4c3r8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33382",
  "fulltext_url": null,
  "id": "5kstzzfr4c3r8",
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  "landing_url": "https://arxiv.org/abs/2610.33382",
  "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.QA"
    ],
    [
      "openalex",
      "2602|Advanced Topics in Algebra"
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  "openalex": "W7223922003",
  "pdf_url": "https://arxiv.org/pdf/2610.33382",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5kstzzfr4c3r8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.33382",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33382"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Classical cumulants from the double tensor unshuffle Hopf algebra",
  "title_text": "Classical cumulants from the double tensor unshuffle Hopf algebra",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/5kstzzfr4c3r8",
  "venue": "arXiv",
  "versions": []
}