Robust Cycles in a Random Permutation

Let $\sigma$ be a uniformly random permutation of $\{1, 2, \dots, n\}$. Find the expected number of cycles in the cycle decomposition of $\sigma$. Express your answer as a familiar function of $n$.

Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.