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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


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 lmgtfy.com/?q=n-cycle+permutation&l=1 –  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... –  Guess who it is. Sep 30 '11 at 7:23

1 Answer 1

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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