# Permutations from given length and cycle

If you know the length $n$ of a permutation $p$ as well as the number of its $k$ cycles, can you figure out from this information if $p$ is an even or odd permutation?

My answer: Yes, given the length and number of cycles one can find out if it is even or odd by using the sign of permutations: $(-1)^{n-k}$

Is this correct? Or have I confused it with something else? I'm not very good at this topic of cycles and permutations.

## 1 Answer

if you have a cycle of lenght $m$ it's sign is given by $(-1)^{m-1}$. This can be shown with the following $$(1,2,\ldots,m) = (1,2)(1,3)\ldots(1,m)$$ so if you have $k$ cycles, each one with lengh $m_j$, the sign would be

$$(-1)^{\sum_{j=1}^k (m_j - 1)} = (-1)^{n-k}$$

Where $n = \sum_{j=1}^k m_j$ since it is known that each permutation can be decomposed in the composition of cycles