# Bell Numbers (Number of ways to Partition a Set)

Hello Everyone,

Given a set of n elements, find number of ways of partitioning it.
Examples:

Input: n = 2 Output: Number of ways = 2 Explanation: Let the set be {1, 2} { {1}, {2} } { {1, 2} } Input: n = 3 Output: Number of ways = 5 Explanation: Let the set be {1, 2, 3} { {1}, {2}, {3} } { {1}, {2, 3} } { {2}, {1, 3} } { {3}, {1, 2} } { {1, 2, 3} }.

What is a Bell Number?
Let S(n, k) be total number of partitions of n elements into k sets. The value of n’th Bell Number is sum of S(n, k) for k = 1 to n.
Value of S(n, k) can be defined recursively as, S(n+1, k) = k*S(n, k) + S(n, k-1)
How does above recursive formula work?
When we add a (n+1)’th element to k partitions, there are two possibilities.

1. It is added as a single element set to existing partitions, i.e, S(n, k-1)
2. It is added to all sets of every partition, i.e., k*S(n, k)
S(n, k) is called Stirling numbers of the second kind
First few Bell numbers are 1, 1, 2, 5, 15, 52, 203, ….
A Simple Method to compute n’th Bell Number is to one by one compute S(n, k) for k = 1 to n and return sum of all computed values.
A Better Method is to use Bell Triangle. Below is a sample Bell Triangle for first few Bell Numbers.

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

The triangle is constructed using below formula.

// If this is first column of current row ‘i’ If j == 0 // Then copy last entry of previous row // Note that i’th row has i entries Bell(i, j) = Bell(i-1, i-1) // If this is not first column of current row Else // Then this element is sum of previous element // in current row and the element just above the // previous element Bell(i, j) = Bell(i-1, j-1) + Bell(i, j-1)

Interpretation
Then Bell(n, k) counts the number of partitions of the set {1, 2, …, n + 1} in which the element k + 1 is the largest element that can be alone in its set.
For example, Bell(3, 2) is 3, it is count of number of partitions of {1, 2, 3, 4} in which 3 is the largest singleton element. There are three such partitions:

{1}, {2, 4}, {3} {1, 4}, {2}, {3} {1, 2, 4}, {3}.

Below is Dynamic Programming based implementation of above recursive formula.

`// A C++ program to find n'th Bell number`

`#include<iostream>`

`using` `namespace` `std;`

`int` `bellNumber(` `int` `n)`

`{`

` ` `int` `bell[n+1][n+1];`

` ` `bell[0][0] = 1;`

` ` `for` `(` `int` `i=1; i<=n; i++)`

` ` `{`

` ` `// Explicitly fill for j = 0`

` ` `bell[i][0] = bell[i-1][i-1];`

` ` `// Fill for remaining values of j`

` ` `for` `(` `int` `j=1; j<=i; j++)`

` ` `bell[i][j] = bell[i-1][j-1] + bell[i][j-1];`

` ` `}`

` ` `return` `bell[n][0];`

`}`

`// Driver program`

`int` `main()`

`{`

` ` `for` `(` `int` `n=0; n<=5; n++)`

` ` `cout << ` `"Bell Number "` `<< n << ` `" is "`

` ` `<< bellNumber(n) << endl;`

` ` `return` `0;`

`}`

Output:

Bell Number 0 is 1 Bell Number 1 is 1 Bell Number 2 is 2 Bell Number 3 is 5 Bell Number 4 is 15 Bell Number 5 is 52

Time Complexity of above solution is O(n2). We will soon be discussing other more efficient methods of computing Bell Numbers.