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) < cnn!√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.

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