Question 77
Given an integer number $n \ge 2$, evaluate the sum
$$
\sum_{\sigma \in S_n} (\operatorname{sgn} \sigma) n^{\ell(\sigma)},
$$
where $S_n$ is the set of all $n$-element permutations, and $\ell(\sigma)$ is the number of disjoint cycles in the standard decomposition of $\sigma$.
$$
\sum_{\sigma \in S_n} (\operatorname{sgn} \sigma) n^{\ell(\sigma)},
$$
where $S_n$ is the set of all $n$-element permutations, and $\ell(\sigma)$ is the number of disjoint cycles in the standard decomposition of $\sigma$.