1. How many cyclic permutations are there of set with n elements

2. Permutation$\displaystyle \pi e S_{n}$ has k cycles of powers $\displaystyle c_{1}..c_{k}$

show that: a) $\displaystyle sgn(\pi)=\prod^{k}_{i=1} (-1)^{c_{i}-1}$

$\displaystyle b) sgn(\pi)=(-1)^{n-k}$

for 1.. isn't it just simply n! if the cyclic permutation is defined as one with just 1 cycle and $\displaystyle \sum_{i=2}^{n} i!$ if it's defined as one with just one and only cycle that has more than 2 elements in it? how to prove it if it's true?

2. No idea, also power of cycle tells us how many elements each cycle has, right?