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here is such theorem or axiom ,which states that

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

my question is 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?thanks a lot

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The n-cycle that google brought up is completely different to the type you are interested in. – Ragib Zaman Sep 30 '11 at 5:08
Your search result is about chemistry, not mathematics. Instead, try – Dan Brumleve Sep 30 '11 at 5:10
The n-cycle looks like it's simply the permutation that 'cycles' the numbers like so: $$1\to2\to3\to\cdots n\to1.$$ The counting formula sounds like interesting in and of itself, though, so I'd like to know where you read it. – anon Sep 30 '11 at 7:06
I've seen $k$-cycles, sure... – J. M. Sep 30 '11 at 7:23

An $n$-cycle in this context is a particular [cyclic permutation][1] 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.

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