Let $n = 2k$ and suppose that $\sigma$ is a permutation in $S_n$ which is equal to a product of k disjoint cycles of length $2$. In how many ways one can write $\sigma$ as a product of two cycles of length $n$? Clearly when $k$ is odd the number is zero. So we can assume that $k$ is even.

2$\begingroup$ Not sure if the votes to close are justified, looks like a reasonable question to me. Maybe the OP could tell us why he is interested in his question. $\endgroup$ – Peter Mueller Jan 18 '14 at 17:37

1$\begingroup$ It will depend on the cycle type of $\sigma$. You could in principal calculate it for any given cycle type from the character table of $S_n$. (I also don't understand the votes to close.) $\endgroup$ – Derek Holt Jan 18 '14 at 18:30

$\begingroup$ @RichardStanley your post is missing a link :( $\endgroup$ – Igor Rivin Jan 18 '14 at 19:45

$\begingroup$ @Derek Holt: I understand the question that $\sigma$ is a product of $k$ disjoint transpositions. $\endgroup$ – Peter Mueller Jan 18 '14 at 19:50

2$\begingroup$ One place (of several) where this problem is discussed is math.mit.edu/~rstan/pubs/pubfiles/47.pdf. See Theorem 3.1. A further reference (with many additional references) is math.uwaterloo.ca/math/sites/ca.math/files/uploads/files/…. $\endgroup$ – Richard Stanley Jan 18 '14 at 21:38