\(\mathfrak{S}_n\) is amazing (part 2)

See part 1 for the definition of \(\mathfrak{S}_n\) and standard representation.

Let \(\mathbf{E}_n\) denote the expected value of the number of cycles in the standard representation of permutation \(w \in \mathfrak{S}_n\). What is the value of \(\lfloor \mathbf{E}_{1000} \rfloor\)?

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