## Sign of the inverse permutation

Permutations $$p$$ and $$p^{-1}$$ have the same cycle lengths (on the same elements), since $$p^{-1}$$ is obtained from $$p$$ by the reverse of the arrows.
Permutations $$p$$ and $$p^{-1}$$ have the same number of inversions, since in the bipartite graph the crossings remain unchanged if we reverse the arrows.
If $$p=t_1\circ t_2\circ…\circ t_k$$ is a factorization of $$p$$ into transpositions, then $$p^{-1}=t_k\circ t_{k-1}\circ…\circ t_1$$. Observe that the composition yields the identity.
From $$sgn(p)\cdot sgn(p^{-1})=sgn(p\circ p^{-1})=sgn(\imath)=1$$ yields $$sgn(p)= sgn(p^{-1})$$.