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Friday, April 23, 2010

Stirling numbers of the second kind - continued

As we can deduce from the recurrence
 $$C(n,k) = C(n-1,k-1) + C(n-1,k)$$ that
$$C(n,k) = \frac{1}{k!} \prod_{j=0}^{k-1}(n-j),$$
where $C(n,k)$ is the number of k-subsets out of an n-set, we can similarly find a closed form for the recurrence $$S(n,k) = S(n-1,k-1) + k \cdot S(n-1,k)$$ which is
$$S(n,k) = \frac{1}{k!} \sum_{j=0}^{k} (-1)^{k-j} j^n C(k,j) ,$$ where $S(n,k)$ is the number of k-partitions of an n-set, a Stirling number of the second kind.

The tech for doing discrete sums is really spectaculair imho and therefore want to know much more about it.

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