Semi-Latin Squares and Related Designs

Abstract

SUMMARY A semi-Latin square for t = kn symbols is a rectangular arrangement with n rows and t columns, these latter being grouped into sets each containing k consecutive columns; each symbol occurs exactly once in each row and exactly once in each set of columns (Yates, 1935). Previous authors seem not to have considered the combinatorial possibilities of semi-Latin squares. For each pair of values k and t, the semi-Latin squares can be classified into “species” or “main classes”, akin to the species of Latin squares. The classification considered in this note disregards the ordering of the k symbols to be found where any row intersects any set of columns; for k = 2 and t = 4, 6 and 8, the numbers of species are 1, 2 and 10 respectively. The relationship between semi-Latin squares, Trojan squares and certain partially balanced incomplete block designs with two associate classes is discussed. The relevant semi-regular designs of Clatworthy (1973) are examined to see which can be obtained from semi-Latin (Trojan) squares, and some enumerations of squares obtainable from the semi-regular designs are reported.