Sum of sgn of permutation What is the sum of sign of permutation, i.e.
$$
\sum_{\pi \in S_n} \text{sgn}(\pi),
$$
where $\pi$ is the permutation of $S_n$.

If $n>1$, the answer is $0$: $\operatorname{sgn}: S_n \to \{1,-1\}$ is a homomorphism, so its kernel $\operatorname{sgn}^{-1}(1)$ is a normal subgroup of index $2$. Hence $\lvert \operatorname{sgn}^{-1}(1) \rvert = \lvert \operatorname{sgn}^{-1}(-1) \rvert$.
(If $n=1$, the answer is $1$, of course.)
