Pairwise Reflection Symmetry in Generalized Latin 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.

Generalized Latin rectangles extend this to $\lambda n \times n$ matrices with repeated symbols under controlled column 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 $\lambda$ turns out to be remarkably more subtle than for even $\lambda$.

After showing that existence holds also for sufficiently large odd $\lambda$, we initiate the search for the smallest possible value of $\lambda$ in this setting.

We obtain the insight that a column multiplicity of $\lambda=1$ 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.