Difference between revisions of "Bell numbers"
From Encyclopedia of Mathematics
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| − | The Bell numbers | + | {{TEX|done}} |
| + | 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 | or by | ||
| − | + | $$B_{n+1}=\sum_{k=0}^n\binom nkB_k.$$ | |
Also, | Also, | ||
| − | + | $$B_n=\sum_{k=1}^nS(n,k),$$ | |
| − | where | + | 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. |
| − | They are equal to | + | They are equal to $1,1,2,5,15,52,203,877,4140,\ldots$. |
The name honours E.T. Bell. | The name honours E.T. Bell. | ||
Revision as of 14:45, 19 April 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 (cf. Combinatorial analysis) of the second kind, 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