{
  "abstract": "In a Latin square, every row can be interpreted as a permutation, and therefore has a parity (even or odd). We prove that in a uniformly random $n\\times n$ Latin square, the probability that all rows are even is of the form $2^{-n+o(n)}$. This fills a gap left by recent work of Kwan, Petrova and Sawhney: they proved that the row parities behave like a sequence of independent coin flips in a variety of different senses, but they were not able to lower-bound the all-even probability, as their approach had a small number of \"exceptional rows\" whose parities they could not control. In fact, we prove a more general theorem on the probability of any possible prescription of row parities, column parities and symbol parities. Our approach extends the ideas of Kwan, Petrova and Sawhney, introducing a new way to \"bootstrap\" intercalate switchings to control more complex switching operations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13487",
  "author_total": 4,
  "authors": [
    {
      "name": "Catherine Greenhill"
    },
    {
      "name": "Lenka Kopfová"
    },
    {
      "name": "Matthew Kwan"
    },
    {
      "name": "Kalina Petrova"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/78bnvwh3gfcg6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13487",
  "fulltext_url": null,
  "id": "78bnvwh3gfcg6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13487",
  "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",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7221538025",
  "pdf_url": "https://arxiv.org/pdf/2610.13487",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/78bnvwh3gfcg6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.13487",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13487"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On the probability that a random Latin square is all-even",
  "title_text": "On the probability that a random Latin square is all-even",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/78bnvwh3gfcg6",
  "venue": "arXiv",
  "versions": []
}