  In this paper, we enumerate the pairs of permutations that are long cycles
and whose product has a given cycle-type. Our main result is a simple relation
concerning the desired numbers for a few related cycle-types. The relation
refines a formula of the number of pairs of long cycles whose product has $k$
cycles independently obtained by Zagier and Stanley relying on group
characters, and was previously obtained by F\'{e}ray and Vassilieva by counting
some colored permutations first and then relying on some algebraic computations
in the ring of symmetric functions. Our approach here is simpler and
combinatorial.
