1. ## Permutations

1. How many cyclic permutations are there of set with n elements
2. Permutation $\pi e S_{n}$ has k cycles of powers $c_{1}..c_{k}$
show that: a) $sgn(\pi)=\prod^{k}_{i=1} (-1)^{c_{i}-1}$
$b) sgn(\pi)=(-1)^{n-k}$

for 1.. isn't it just simply n! if the cyclic permutation is defined as one with just 1 cycle and $\sum_{i=2}^{n} i!$ 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?

2. ## Re: Permutations

OK so for 1 I know it's (n-1)! (because cycle 1234 is the same as 2341) but I don't know how to prove it

for 2a) we know that if cycle has ck elements then it's sign is (-1)ck-1 and we know that sgn(a*b)=sgn(a)*sgn(b). Does the second equality mean that $sgn(c_{1}* ... * c_{k}) = \prod_{i=1}^{k} sgn(c_{i})$ ? If no, how to prove it?

for 2b) we would have to prove that $\sum_{i=1}^{k} c_{i} = n$ but I don't know how to prove it - help?