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The Bell numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110240/b1102401.png" /> are given by
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The Bell numbers $B_0,B_1,\ldots$ are given by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110240/b1102402.png" /></td> </tr></table>
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$$\sum_{n=0}^\infty B_n\frac{x^n}{n!}=e^{e^x-1}$$
  
 
or by
 
or by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110240/b1102403.png" /></td> </tr></table>
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$$B_{n+1}=\sum_{k=0}^n\binom nkB_k.$$
  
 
Also,
 
Also,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110240/b1102404.png" /></td> </tr></table>
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$$B_n=\sum_{k=1}^nS(n,k),$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110240/b1102405.png" /> are Stirling numbers (cf. [[Combinatorial analysis|Combinatorial analysis]]) of the second kind, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110240/b1102406.png" /> is the total number of partitions of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110240/b1102407.png" />-set.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110240/b1102408.png" />.
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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
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