Difference between revisions of "Bell numbers"
From Encyclopedia of Mathematics
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$$B_n=\sum_{k=1}^nS(n,k),$$ | $$B_n=\sum_{k=1}^nS(n,k),$$ | ||
| − | where $S(n,k)$ are Stirling numbers (cf. [[ | + | 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 $1,1,2,5,15,52,203,877,4140,\ldots$. | They are equal to $1,1,2,5,15,52,203,877,4140,\ldots$. | ||
Revision as of 19:16, 21 November 2014
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=31865
Bell numbers. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bell_numbers&oldid=31865
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