Counting pairs of cycles whose product is a permutation with restricted cycle lengths

We find exact and asymptotic formulas for the number of pairs $(p,q)$ of $N$-cycles such that the all cycles of the product $p\cdot q$ have lengths from a given integer set. We then apply these results to prove a surprisingly high lower bound for the number of permutations whose block transposition distance from the identity is at least $(n+1)/2$.