Enumerative Combinatorics · Set partitions, Stirling numbers, and the cycle structure of permutations
Let S(n,k) count partitions of [n] into k nonempty unordered blocks
Problem
Let \(S(n,k)\) count partitions of \([n]\) into \(k\) nonempty unordered blocks. Derive a recurrence by examining the block containing \(n\). State all initial and out-of-range values needed to determine the array, including the empty-set case. Use the recurrence to construct row \(n=6\) and find \(S(6,3)\). Explain why the factor multiplying \(S(n-1,k)\) is \(k\) even though the blocks themselves are unordered.
Hint
Ask whether \(n\) is alone in its block.
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