The number of inversions in a permutation is equal to the number of its inverse permutation.

Question

I need to prove that the sign of a permutation is equal to the sign of the inverse of the permutation. I understand it is true, but how do you proof that the #inversions of $\sigma$= #inversions of $\sigma^{-1}$? Can anyone help me out? I know that the inversions are not the same, I tried it with an example.

Answer 1

Hint:

$$sgn(\sigma)\cdot sgn(\tau)=sgn(\sigma \tau)$$

Answer 2

Saying $\sigma$ inverts $i$ and $j$ means that $i$ and $j$ come in the opposite order to $\sigma(i)$ and $\sigma(j)$.

Therefore $\sigma$ inverts $i$ and $j$ if and only if $\sigma^{-1}$ inverts $\sigma(i)$ and $\sigma(j)$. This gives a one-to-one correspondence between the inversions of $\sigma$ and $\sigma^{-1}$.