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The Bell numbers $B_0,B_1,\ldots$ are given by
 
The Bell numbers $B_0,B_1,\ldots$ are given by
  
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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. [[Combinatorial analysis|Combinatorial analysis]]) of the second kind, so that $B_n$ is the total number of partitions of an $n$-set.
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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 $1,1,2,5,15,52,203,877,4140,\ldots$.
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They are equal to $1,1,2,5,15,52,203,877,4140,\ldots$ ({{OEIS|A000110}}).
  
 
The name honours E.T. Bell.
 
The name honours E.T. Bell.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Comtet,   "Advanced combinatorics" , Reidel (1974)</TD></TR></table>
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* L. Comtet, "Advanced combinatorics", Reidel (1974) {{ZBL|0283.05001}}

Latest revision as of 07:26, 7 November 2023

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 $1,1,2,5,15,52,203,877,4140,\ldots$ (OEIS sequence A000110).

The name honours E.T. Bell.

References

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