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Here is a certain theorem or axiom, which states the following:

(*) Let $n$ be an odd number. The number of way to write the $n$-cycle $(1,2,\dots,n)$ in the form $uvu^{-1}v^{-1}$, is equal to $2n\cdot n!/(n+1)$.

What is $n$-cycle? When I have tried to search in Google, it said that it is nitrogen cycles, which is defined like this:

The nitrogen cycle is the process by which nitrogen is converted between its various chemical forms.

Is it so? And what kind of application it has in numbers and permutations?

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    I've seen [$k$-cycles](http://en.wikipedia.org/wiki/Cycle_%28mathematics%29), sure...2011-09-30

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An $n$-cycle in this context is a particular cyclic permutation of length $n$ of $n$ elements, compare the comments to your post.

But note that the formula in your OP cannot be correct, because for $n+1$ an odd prime it does not give an integer.