# Bell Numbers

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Bell Numbers, or Bell's Numbers counts the number of ways N objects can be partitioned into groups that are non-empty.

${\displaystyle B_{n}}$ can be defined as the following:

${\displaystyle B_{n}=\Sigma _{k=0}^{n}S(n,k)}$

where ${\displaystyle S(n,k)}$ is the Stirling Number of the Second Kind.

## Bell's Triangle

Bell Numbers can also be calculated using Bell's Triangle.

1
1 2
2 3 5
5 7 10 15
15 20 27 37 52


The concept behind this is that it can be calculated with only addition - the first column of a number is equal to the last value of the previous row, and subsequential columns' values are from the sum of the value immediately preceding it, and the value on top of that value.

It can also be viewed as a recurrence (${\displaystyle i}$-th row, ${\displaystyle j}$-column)

• ${\displaystyle B(0,0)=1}$
• ${\displaystyle B(i,0)=B(i-1,i-1)}$
• ${\displaystyle B(i,j)=B(i,j-1)+B(i-1,j-1),0

where the sum of row ${\displaystyle i}$ is ${\displaystyle i}$-th Bell Number.