Pairwise Reflection Symmetry in Frequency Rectangles
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,…,n} appears once per row and column of an n× n matrix. Frequency rectangles extend this to λn × 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.
Related
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BibTeX
@misc{iurlano2026pairwise,
title = {{Pairwise Reflection Symmetry in Frequency Rectangles}},
author = {Enrico Iurlano and Günther R. Raidl},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2606.28315},
eprint = {2606.28315},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.28315},
}
RIS
TY - UNPB
TI - Pairwise Reflection Symmetry in Frequency Rectangles
AU - Enrico Iurlano
AU - Günther R. Raidl
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2606.28315
UR - https://arxiv.org/abs/2606.28315
AB - 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,…,n} appears once per row and column of an n× n matrix. Frequency rectangles extend this to λn × 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.
ER -