Given $S = \{1, 2, \dots, 2n\}$, we can always pair off elements into $n$ pairs such that each sum to a prime. The proof of this fact is easy and follows from Bertrand's postulate.

Now, let $\gamma(n)$ be the number of different ways to pair off the set $S$ into pairs that sum to primes. What is known of the asymptotics of this function?

From computer simulations, the first couple values are $1, 2, 3, 6, 26, 96, 210, 1106, 3759, 12577, 74072, 423884$ so it seems to grow fairly quickly.