Namespaces
Variants
Actions

Difference between revisions of "Bell numbers"

From Encyclopedia of Mathematics
Jump to: navigation, search
(TeX)
m (link)
Line 12: Line 12:
 
$$B_n=\sum_{k=1}^nS(n,k),$$
 
$$B_n=\sum_{k=1}^nS(n,k),$$
  
where $S(n,k)$ are Stirling numbers (cf. [[Combinatorial analysis|Combinatorial analysis]]) of the second kind, so that $B_n$ is the total number of partitions of an $n$-set.
+
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
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