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

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} }.


Solution to above questions is Bell Number.

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. Refer this for computation of S(n, k).

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.

## C++

 // A C++ program to find n'th Bell number  #include  using namespace std;     int bellNumber(int n)  {     int bell[n+1][n+1];     bell = 1;     for (int i=1; i<=n; i++)     {        // Explicitly fill for j = 0        bell[i] = 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];  }     // Driver program  int main()  {     for (int n=0; n<=5; n++)        cout << "Bell Number " << n << " is "             << bellNumber(n) << endl;     return 0;  }

## Java

 // Java program to find n'th Bell number  import java.io.*;     class GFG   {      // Function to find n'th Bell Number      static int bellNumber(int n)      {          int[][] bell = new int[n+1][n+1];          bell = 1;                     for (int i=1; i<=n; i++)          {              // Explicitly fill for j = 0              bell[i] = 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];      }             // Driver program      public static void main (String[] args)       {          for (int n=0; n<=5; n++)              System.out.println("Bell Number "+ n +                              " is "+bellNumber(n));      }  }     // This code is contributed by Pramod Kumar

## Python3

 # A Python program to find n'th Bell number     def bellNumber(n):         bell = [[0 for i in range(n+1)] for j in range(n+1)]      bell = 1     for i in range(1, n+1):             # Explicitly fill for j = 0          bell[i] = bell[i-1][i-1]             # Fill for remaining values of j          for j in range(1, i+1):              bell[i][j] = bell[i-1][j-1] + bell[i][j-1]         return bell[n]     # Driver program  for n in range(6):      print('Bell Number', n, 'is', bellNumber(n))     # This code is contributed by Soumen Ghosh

## C#

 // C# program to find n'th Bell number  using System;     class GFG {             // Function to find n'th       // Bell Number      static int bellNumber(int n)      {          int[,] bell = new int[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 Code      public static void Main ()       {          for (int n = 0; n <= 5; n++)              Console.WriteLine("Bell Number "+ n +                                " is "+bellNumber(n));      }  }     // This code is contributed by nitin mittal.

## PHP

 

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.

Another problem that can be solved by Bell Numbers.
A number is squarefree if it is not divisible by a perfect square other than 1. For example, 6 is a square free number but 12 is not as it is divisible by 4.
Given a squarefree number x, find the number of different multiplicative partitions of x. The number of multiplicative partitions is Bell(n) where n is number of prime factors of x. For example x = 30, there are 3 prime factors of 2, 3 and 5. So the answer is Bell(3) which is 5. The 5 partitions are 1 x 30, 2 x15, 3 x 10, 5 x 6 and 2 x 3 x 5.

Exercise:
The above implementation causes arithmetic overflow for slightly larger values of n. Extend the above program so that results are computed under modulo 1000000007 to avoid overflows.