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?