On the number of symmetric Latin squares

The number of symmetric Latin squares is closely related with the security of the post-commutative quasigroups cipher. Let L n denote the number of distinct n×n Latin squares. It is fairly well known that (n!) 2n n-n 2 ≤ L n ≤ π k=1 n (k!)n/k, and asymptotically L n ~ (n/e)n 2 as → ∞. Let S n denote the number of distinct n×n symmetric Latin squares. In this paper, we give a lower bound of S n and show that S n ~ n! · n 3 / 8 n 2 (n → ∞) when n is odd, and S n ~ n! · (n-1)!!·n 3 / 8 n 2 (n → ∞) when n is even.