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.
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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 MuellerJan 18, 2014 at 17:37
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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 HoltJan 18, 2014 at 18:30
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$\begingroup$ @RichardStanley your post is missing a link :( $\endgroup$– Igor RivinJan 18, 2014 at 19:45
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$\begingroup$ @Derek Holt: I understand the question that $\sigma$ is a product of $k$ disjoint transpositions. $\endgroup$– Peter MuellerJan 18, 2014 at 19:50
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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 StanleyJan 18, 2014 at 21:38
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