{
  "abstract": "Many combinatorial designs ask for equal distribution of given symbols across the entries of a matrix. The paramount examples are Latin squares, where each symbol from $\\{1,\\dots,n\\}$ appears once per row and column of an $n\\times n$ matrix. Frequency rectangles extend this to $λn \\times μn$ matrices with repeated symbols under controlled column and row frequencies. In this more general setting, we examine structural properties of pairwise reflection-symmetry, which requires that, on every pair of columns, each ordered symbol pair $(p,q)$ occurs as often as its reversal $(q,p)$. This order-balance is precisely what makes head-to-head comparisons unbiased, i.e., no symbol gains a systematic advantage from the position it occupies relative to another, a fairness demand arising for instance when scheduling tournaments or laying out comparative trials. Existence of such objects for odd $λ$ turns out to be remarkably more subtle than for even $λ$. After showing that existence holds also for sufficiently large odd $λ$, we initiate the search for the smallest possible value of $λ$ in this setting. We obtain the insight that a per-column multiplicity of $λ=1$ can be achieved if and only if $n$ is a power of two. We complement the existence results with a direct product construction and add several further observations on the property. Finally, we propose and evaluate a quadratically constrained integer program to computationally search for these objects. The resulting experiments reveal that many of them possess an underlying group-theoretic structure which, as we conjecture, may even be unavoidable in certain settings.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.28315",
  "author_total": 2,
  "authors": [
    {
      "name": "Enrico Iurlano"
    },
    {
      "name": "Günther R. Raidl"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6pg5dn314v8tm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.28315",
  "fulltext_url": null,
  "id": "6pg5dn314v8tm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.28315",
  "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",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7166539629",
  "pdf_url": "https://arxiv.org/pdf/2606.28315",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6pg5dn314v8tm.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2606.28315",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.28315"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Pairwise Reflection Symmetry in Frequency Rectangles",
  "title_text": "Pairwise Reflection Symmetry in Frequency Rectangles",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:25Z",
  "url": "https://research.draconitic.com/paper/6pg5dn314v8tm",
  "venue": "arXiv",
  "versions": []
}