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

Recommended practice

The 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).
Below is Dynamic Programming based implementation of the above recursive code using the Stirling number-

## C++

 #include  using namespace std;   int main() {     int n=5;     int s[n+1][n+1];     for(int i=0;ii) s[i][j]=0;             else if(i==j) s[i][j]=1;             else if(i==0 || j==0) s[i][j]=0;             else{                                   s[i][j]= j*s[i-1][j] + s[i-1][j-1];             }                       }     }     int ans=0;     for(int i=0;i

## Java

 /*package whatever //do not write package name here */ // Java program to find number of ways of partitioning it.   import java.io.*; // "static void main" must be defined in a public class. public class GFG {     public static void main(String[] args)     {         int n = 5;         int[][] s = new int[n + 1][n + 1];         for (int i = 0; i < n + 1; i++) {             for (int j = 0; j < n + 1; j++) {                 if (j > i)                     s[i][j] = 0;                 else if (i == j)                     s[i][j] = 1;                 else if (i == 0 || j == 0)                     s[i][j] = 0;                 else {                     s[i][j]                         = j * s[i - 1][j] + s[i - 1][j - 1];                 }             }         }         int ans = 0;         for (int i = 0; i < n + 1; i++) {             ans += s[n][i];         }         System.out.println(ans);     } }   // The code is contributed by Gautam goel (gautamgoel962)

## Python3

 # python program to find number of ways of partitioning it. n = 5 s = [[0 for _ in range(n+1)] for _ in range(n+1)] for i in range(n+1):     for j in range(n+1):         if j > i:             continue         elif(i==j):             s[i][j] = 1         elif(i==0 or j==0):             s[i][j]=0         else:             s[i][j] = j*s[i-1][j] + s[i-1][j-1] ans = 0 for i in range(0,n+1):     ans+=s[n][i] print(ans)

## C#

 // C# Program to find number of ways of partitioning it. using System;   public class Program {     static public void Main(string[] args) {           int n = 5;           int[, ] s = new int[n + 1, n + 1];           for (int i = 0; i < n + 1; i++) {               for (int j = 0; j < n + 1; j++) {                   if (j > i)                     s[i, j] = 0;                   else if (i == j)                     s[i, j] = 1;                   else if (i == 0 || j == 0)                     s[i, j] = 0;                   else                     s[i, j]                         = j * s[i - 1, j] + s[i - 1, j - 1];             }         }           int ans = 0;           for (int i = 0; i < n + 1; i++)             ans += s[n, i];           Console.WriteLine(ans);     } }   // This code is contributed by Tapesh(tapeshdua420)

## Javascript

 // JavaScript program to find number of ways of partitioning it.   let n=5; let s = new Array(n+1); for(let i=0;ii) s[i][j]=0;         else if(i==j) s[i][j]=1;         else if(i==0 || j==0) s[i][j]=0;         else{               s[i][j]= j*s[i-1][j] + s[i-1][j-1];         }       } } let ans=0; for(let i=0;i

Output

52



Time complexity: O(N2
Auxiliary Space: O(N2

A Better Method is to use Bell Triangle. Below is a sample Bell Triangle for first few Bell Numbers.

11 22 3 55 7 10 1515 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 rowElse    // 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++14

 // A C++ program to find n'th Bell number #include 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; }

## 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[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     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[0][0] = 1     for i in range(1, n+1):           # Explicitly fill for j = 0         bell[i][0] = 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][0]   # 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.

## Javascript

 

## 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: O(N2
Auxiliary Space: O(N2)

Space Optimized DP Approach:

We can use a 1-D list to represent the previous row of the Bell triangle. We initialize dp[0] to 1, since there is only one way to partition an empty set.

To compute the Bell numbers for n > 0, we first set dp[0] = dp[i-1], since the first element in each row is the same as the last element in the previous row. Then, we use the recurrence relation dp[j] = prev + dp[j-1] to compute the Bell number for each partition, where prev is the value of dp[j] in the previous iteration of the inner loop. We update prev to the temporary variable temp before updating dp[j].

Finally, we return dp[0], which is the Bell number for the partition of a set with n elements into non-empty subsets.

## C++

 #include  #include    // Function to calculate the Bell number for 'n' int bellNumbers(int n) {     // Initialize the previous row of the Bell triangle with     // dp[0] = 1     std::vector<int> dp(n + 1, 0);     dp[0] = 1;       for (int i = 1; i <= n; i++) {         // The first element in each row is the same as the         // last element in the previous row         int prev = dp[0];         dp[0] = dp[i - 1];         for (int j = 1; j <= i; j++) {             // The Bell number for n is the sum of the Bell             // numbers for all previous partitions             int temp = dp[j];             dp[j] = prev + dp[j - 1];             prev = temp;         }     }       return dp[0]; }   int main() {     int n = 5;     std::cout << bellNumbers(n) << std::endl;       return 0; }

## Java

 import java.util.Arrays;   public class BellNumbers {       // Function to calculate the Bell number for 'n'     static int bellNumbers(int n)     {         // Initialize the previous row of the Bell triangle         // with dp[0] = 1         int[] dp = new int[n + 1];         Arrays.fill(dp, 0);         dp[0] = 1;           for (int i = 1; i <= n; i++) {             // The first element in each row is the same as             // the last element in the previous row             int prev = dp[0];             dp[0] = dp[i - 1];               for (int j = 1; j <= i; j++) {                 // The Bell number for n is the sum of the                 // Bell numbers for all previous partitions                 int temp = dp[j];                 dp[j] = prev + dp[j - 1];                 prev = temp;             }         }           return dp[0];     }       public static void main(String[] args)     {         int n = 5;         System.out.println(bellNumbers(n));     } }

## Python3

 def bell_numbers(n):     # Initialize the previous row of the Bell triangle with dp[0] = 1     dp = [1] + [0] * n       for i in range(1, n+1):         # The first element in each row is the same as the last element in the previous row         prev = dp[0]         dp[0] = dp[i-1]         for j in range(1, i+1):             # The Bell number for n is the sum of the Bell numbers for all previous partitions             temp = dp[j]             dp[j] = prev + dp[j-1]             prev = temp       return dp[0]     n = 5 print(bell_numbers(n))

## C#

 using System;   class Program {     // Function to calculate the Bell number for 'n'     static int BellNumbers(int n)     {         // Initialize the previous row of the Bell triangle         // with dp[0] = 1         int[] dp = new int[n + 1];         dp[0] = 1;           for (int i = 1; i <= n; i++) {             // The first element in each row is the same as             // the last element in the previous row             int prev = dp[0];             dp[0] = dp[i - 1];               for (int j = 1; j <= i; j++) {                 // The Bell number for n is the sum of the                 // Bell numbers for all previous partitions                 int temp = dp[j];                 dp[j] = prev + dp[j - 1];                 prev = temp;             }         }           return dp[0];     }       static void Main()     {         int n = 5;         Console.WriteLine(BellNumbers(n));     } }

Output

52



Time Complexity:
Auxiliary Space:

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.
Reference:
https://en.wikipedia.org/wiki/Bell_number
https://en.wikipedia.org/wiki/Bell_triangle