Bell numbers

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2010 Mathematics Subject Classification: Primary: 11B73 [MSN][ZBL]

The Bell numbers $B_0,B_1,\ldots$ are given by

$$\sum_{n=0}^\infty B_n\frac{x^n}{n!}=e^{e^x-1}$$

or by

$$B_{n+1}=\sum_{k=0}^n\binom nkB_k.$$

Also,

$$B_n=\sum_{k=1}^nS(n,k),$$

where $S(n,k)$ are Stirling numbers of the second kind (cf. Combinatorial analysis), so that $B_n$ is the total number of partitions of an $n$-set.

They are equal to $1,1,2,5,15,52,203,877,4140,\ldots$.

The name honours E.T. Bell.

References

 [a1] L. Comtet, "Advanced combinatorics" , Reidel (1974)
How to Cite This Entry:
Bell numbers. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bell_numbers&oldid=35757
This article was adapted from an original article by N.J.A. Sloane (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article