Uniform semi-Latin squares and their Schur-optimality

Let n and k be integers, with n > 1 and k > 0. An (n×n)/k semiLatin square S is an n × n array, whose entries are k-subsets of an nk-set, the set of symbols of S, such that each symbol of S is in exactly one entry in each row and exactly one entry in each column of S. SemiLatin squares form an interesting class of combinatorial objects which are useful in the design of comparative experiments. We say that an (n × n)/k semi-Latin square S is uniform if there is a constant μ such that any two entries of S, not in the same row or column, intersect in exactly μ symbols (in which case k = μ(n− 1)). We prove that a uniform (n × n)/k semi-Latin square is Schur-optimal in the class of (n×n)/k semi-Latin squares, and so is optimal (for use as an experimental design) with respect to a very wide range of statistical optimality criteria. We give a simple construction to make an (n×n)/k semi-Latin square S from a transitive permutation group G of degree n and order nk, and show how certain properties of S can be determined from permutation group properties of G. If G is 2-transitive then S is uniform, and this provides us with Schur-optimal semi-Latin squares for many values of n and k for which optimal (n × n)/k semi-Latin squares were previously unknown for any optimality criterion. The existence of a uniform (n × n)/((n − 1)μ) semi-Latin square for all