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## Bell numbers

Definition 8.6.12   The Bell number is the number of partitions of .

It is obvious from the definitions that .

Lemma 8.6.13

Proof. For, put the element in with a -subset of for to .

There isn't a nice closed formula for , but there is a nice expression for its exponential generating function.

Definition 8.6.14   The exponential generating function that is associated with the sequence is

If we have and (with obvious notation) and then , the exponential convolution of and . Hence is the coefficient of in the exponential convolution of the sequences and . Thus . (Shifting is achieved by differentiation for exponential generating functions.) Therefore and using the condition we find that . So

root 2002-06-10