{
  "abstract": "We introduce a new class of unitary set-theoretic Yang--Baxter solutions, called \\emph{bipartite} solutions. This class, which is purely combinatorial in nature, forms a full subcategory of $\\mathcal{LYB}$, the category of Lyubashenko type solutions of Agore, Chirvasitu and the author, and is particularly amenable to explicit classification results. We give an intrinsic necessary and sufficient condition for a solution of the Yang--Baxter equation to be isomorphic to a bipartite solution, we settle the isomorphism problem within this class and, as an application, we determine their automorphism groups. If $X$ is a finite set with $|X| = n$, the number of isomorphism classes of bipartite solutions on $X$ is $\\lfloor n/2 \\rfloor + 1 + \\sum_{m=2}^{n} \\bigl( o(m, n-m) - 1 \\bigr)$, where $o(m, k)$ denotes the number of orbits of the symmetric group $S_m$ acting by conjugation on the $k$-element multisets of elements of $S_m$. The paper also offers a comparative study: for small $|X| = n$ we count, up to isomorphism, all set-theoretic Yang--Baxter solutions on $X$, together with five distinguished families of solutions. Throughout the range we were able to compute, the class $\\mathcal{LYB}$ accounts for a large share of all bijective and of all non-degenerate solutions, strikingly close to the total.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.30845",
  "author_total": 1,
  "authors": [
    {
      "name": "Gigel Militaru"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/16bhdc3tgxb5g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.30845",
  "fulltext_url": null,
  "id": "16bhdc3tgxb5g",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.30845",
  "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.QA"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2602|Advanced Topics in Algebra"
    ]
  ],
  "openalex": "W7223340224",
  "pdf_url": "https://arxiv.org/pdf/2610.30845",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/16bhdc3tgxb5g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.30845",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.30845"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Bipartite solutions of the set-theoretic Yang--Baxer equation",
  "title_text": "Bipartite solutions of the set-theoretic Yang--Baxer equation",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/16bhdc3tgxb5g",
  "venue": "arXiv",
  "versions": []
}