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Bell number (Definition)

The Bell number, denoted $ B(n)$ is the total number of partitions of a set with $ n$ elements. For $ n=0$, we have $ B(0)=1$. For $ n\ge 1$, we have

$\displaystyle B(n) = \sum_{k=0}^n S(n,k) \qquad \textrm{for } n \ge 1 $
where $ S(n,k)$ are the Stirling numbers of the second kind.
Proposition 1  
$\displaystyle B(n+1) = \sum_{k=0}^n \binom{n}{k} B(k) $
Proof. We count the number of partitions of a set of $ n+1$ elements, depending on the size of the block containing the $ n+1$st element. If the block has size $ j$ for $ 1 \le j \le n+1$ then we have $ \binom{n}{j-1}$ choices for the $ n$ other elements of the block. The remaining $ n+1=j$ elements can be partitioned in $ B(n+1-j)$ ways. We have therefore that:
$\displaystyle B(n+1)$ $\displaystyle =$ $\displaystyle \sum_{j=1}^{n+1}\binom{n}{j-1} B(n+1-j)$  
  $\displaystyle =$ $\displaystyle \sum_{j=1}^{n+1}\binom{n}{n+1-j} B(n+1-j)$  
  $\displaystyle =$ $\displaystyle \sum_{k=0}^n \binom{n}{k} B(k)$  

$ \qedsymbol$

Using the formula above, one can easily derive the first few Bell numbers. Starting with $ n=0$, the first ten Bell numbers are 1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147.



"Bell number" is owned by aoh45. [ full author list (2) ]
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See Also: Stirling numbers of the second kind


Attachments:
Bell's triangle (Data Structure) by PrimeFan
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Cross-references: block, size, Stirling numbers of the second kind, partitions, number
There is 1 reference to this entry.

This is version 3 of Bell number, born on 2004-11-01, modified 2008-07-14.
Object id is 6436, canonical name is BellNumber.
Accessed 4851 times total.

Classification:
AMS MSC: 11B73 (Number theory :: Sequences and sets :: Bell and Stirling numbers)
 05A18 (Combinatorics :: Enumerative combinatorics :: Partitions of sets)
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