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 equation and asymptotically L
n
∼ (n over e)
ⁿ²
as n ₒ ∞. 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 equation (n → ∞) when n is odd, and equation (n → ∞) when n is even.