Surjective function count

Count the number of surjective (onto) functions from a set of n distinct elements to a set of k distinct labels — i.e., colorings of n items with exactly k colors where every color is used at least once. Return the count as an exact integer (Python big ints). Constraints: 0 <= n <= 200, 0 <= k <= 200. Use inclusion-exclusion: sum over i of (-1)^i * C(k,i) * (k-i)^n.