Sign of the inverse permutation
Task number: 2457
Show four different arguments why the inverse permutatin has the same sign as the original one.
Resolution
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 \(sgn(p)= sgn(p^{-1})\).
