Difference between revisions of "Bell numbers"
From Encyclopedia of Mathematics
(TeX) |
(MSC 11B73) |
||
| (One intermediate revision by the same user not shown) | |||
| Line 1: | Line 1: | ||
| − | {{TEX|done}} | + | {{TEX|done}}{{MSC|11B73}} |
| + | |||
The Bell numbers $B_0,B_1,\ldots$ are given by | The Bell numbers $B_0,B_1,\ldots$ are given by | ||
| Line 12: | Line 13: | ||
$$B_n=\sum_{k=1}^nS(n,k),$$ | $$B_n=\sum_{k=1}^nS(n,k),$$ | ||
| − | 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 18:16, 20 December 2014
2020 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=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