Math 2.0

Number theory Definition

Bell numbers

Builds on Stirling numbers of the second kind

Definition

Let nn be a natural number. The Bell number BnB_n is the number of partitions of an nn-element set into nonempty blocks. Equivalently, it is the sum of Stirling numbers of the second kind: Bn=∑k=0nS(n,k).B_n = \sum_{k=0}^{n} S(n, k).

Remarks

The first values, for n=0,1,…,6n = 0, 1, \dots, 6, are 1,1,2,5,15,52,2031, 1, 2, 5, 15, 52, 203. B0=1B_0 = 1, because the empty set has exactly one partition, the partition with no blocks. The values satisfy a recurrence with binomial coefficients.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Bell_number, Set partitions
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