← 研踪 PaperTrailarXiv:2606.28315 · v2
Pairwise Reflection Symmetry in Frequency Rectangles
首次提交 2026-06-27 01:57
版本更新 2026-11-04 06:17
雷达观测 2026-11-05 11:11
摘要
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.
作者备注：18 pages, 2 figures; Julia source code provided as ancillary file
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v2 · 2026-11-04 06:17 · 最新观测版本
Pairwise Reflection Symmetry in Frequency Rectangles
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.
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