Question 90
Given a permutation $\sigma$ of $\{1,2, \ldots, 2013\}$, let $f(\sigma)$ be the number of fixed points of $\sigma$—that is, the number of $k \in \{1,2, \ldots, 2013\}$ such that $\sigma(k) = k$. If $S$ is the set of all possible permutations $\sigma$, compute
$$
\sum_{\sigma \in S} f(\sigma)^4
$$
(Here, a permutation $\sigma$ is a bijective mapping from $\{1,2, \ldots, 2013\}$ to $\{1,2, \ldots, 2013\}$.)
$$
\sum_{\sigma \in S} f(\sigma)^4
$$
(Here, a permutation $\sigma$ is a bijective mapping from $\{1,2, \ldots, 2013\}$ to $\{1,2, \ldots, 2013\}$.)