I define the sign of a permutation as the number of transpositions that are in any re-writing of $\sigma$ in a product of transpositions. The number of inversions of a permutation $\sigma$ of $S_n$ is the number of pairs that verifies $\{(i,j):~i<j,~\sigma(j)<\sigma(i)\}$.
Now, why are related both definitions? If I know the number of inversions, why do I know the sign of the permutation?
Also, do you know any place where I can find any nice proof that the sign, as I define it, is well defined?
Thank you so much!