{
  "abstract": "Motivated by recent work on $λ$-fold near-factorizations by Kreher, Li, and Stinson (Designs, Codes and Cryptography, 2026), we introduce and systematically investigate $(λ,μ)$-fold factorizations as combinatorial structures with prescribed representation multiplicities. Such a factorization consists of two subsets of a finite group whose products represent each nonidentity element exactly $λ$ times and the identity exactly $μ$ times. These structures generalize $λ$-fold near-factorizations and provide a common framework for several families arising in combinatorial design theory, including strong external difference families with two equal-sized subsets, as well as binary sequence pairs with two-level correlation. Their associated $\\{0,1\\}$-matrices satisfy the Bridges-Ryser equation $XY=λ(J-I)+μI$, linking the factorization problem to a classical matrix framework for combinatorial designs. We establish the foundational theory of $(λ,μ)$-fold factorizations, provide two general constructions, and prove several nonexistence results for finite abelian groups.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15826",
  "author_total": 3,
  "authors": [
    {
      "name": "Donald L. Kreher"
    },
    {
      "name": "Shuxing Li"
    },
    {
      "name": "Douglas R. Stinson"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ac685mtfxn53p.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15826",
  "fulltext_url": null,
  "id": "ac685mtfxn53p",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15826",
  "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.GR"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7221792581",
  "pdf_url": "https://arxiv.org/pdf/2610.15826",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ac685mtfxn53p.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.15826",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15826"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The theory of $(λ,μ)$-fold factorizations of groups",
  "title_text": "The theory of (λ,μ)-fold factorizations of groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/ac685mtfxn53p",
  "venue": "arXiv",
  "versions": []
}