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!)²ⁿn-n² ≤ L_n ≤ π_k=1ⁿ (k!)n/k, and asymptotically L_n ~ (n/e)n² 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! · n³/₈n² (n → ∞) when n is odd, and S_n ~ n! · (n-1)!!·n³/₈n² (n → ∞) when n is even.