The number of transversals in a Latin square

A Latin square of order n is an n × n array of n symbols, in which each symbol occurs exactly once in each row and column. A transversal is a set of n entries, one selected from each row and each column of a Latin square of order n such that no two entries contain the same symbol. Define T (n) to be the maximum number of transversals over all Latin squares of order n. We show that bn ≤ T (n) ≤ cn√n n! for n ≥ 5, where b ≈ 1.719 and c ≈ 0.614. A corollary of this result is an upper bound on the number of placements of n non-attacking queens on an n × n toroidal chess board. Some divisibility properties of the number of transversals in Latin squares based on finite groups are established. We also provide data from a computer enumeration of transversals in all Latin squares of order at most 9, all groups of order at most 23 and all possible turn-squares of order 14.