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Let $P_n$ denote the set of pairs $(x,y)$ of permutations on $S_{2n}$, where each permutation is a product of $n$ disjoint cycles of length two. Let i and j be two fixed elements of the set $\{1,2, \cdots,2n\}$. Select an element $(x,y)$ of $P_n$. What is the probability that the product $xy$ contains $i$ and $j$ in the same cycle?

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    Crosspost: http://mathoverflow.net/questions/110449/combinatorial-properties-of-permutation-groups2012-10-23
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    (The crosspost has been closed in the meantime.)2012-10-23

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