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Jul 10, 2020 at 21:07 comment added Richard Stanley There is some information related to this problem in Exercise 7.68 of Enumerative Combinatorics, vol. 2. For instance, let $n$ be odd. Then the number of ways of writing a fixed $n$-cycle in the form $uvu^{-1}v^{-1}$ is $2n\cdot n!/(n+1)$.
Jul 10, 2020 at 20:14 history answered Per Alexandersson CC BY-SA 4.0