# 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<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;` `}` |

## 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.` |

## PHP

`<?php` `// A PHP program to find` `// n'th Bell number` `// function that returns` `// n'th bell number` `function` `bellNumber(` `$n` `)` `{` ` ` `$bell` `[0][0] = 1;` ` ` `for` `(` `$i` `= 1; ` `$i` `<= ` `$n` `; ` `$i` `++)` ` ` `{` ` ` ` ` `// Explicitly fill for j = 0` ` ` `$bell` `[` `$i` `][0] = ` `$bell` `[` `$i` `- 1]` ` ` `[` `$i` `- 1];` ` ` ` ` `// Fill for remaining` ` ` `// values of j` ` ` `for` `(` `$j` `= 1; ` `$j` `<= ` `$i` `; ` `$j` `++)` ` ` `$bell` `[` `$i` `][` `$j` `] = ` `$bell` `[` `$i` `- 1][` `$j` `- 1] +` ` ` `$bell` `[` `$i` `][` `$j` `- 1];` ` ` `}` ` ` `return` `$bell` `[` `$n` `][0];` `}` `// Driver Code` `for` `(` `$n` `= 0; ` `$n` `<= 5; ` `$n` `++)` `echo` `(` `"Bell Number "` `. ` `$n` `. ` `" is "` ` ` `. bellNumber(` `$n` `) . ` `"\n"` `);` `// This code is contributed by Ajit.` `?>` |

## Javascript

`<script>` ` ` `// Javascript program to find n'th Bell number` ` ` ` ` `// Function to find n'th Bell Number` ` ` `function` `bellNumber(n)` ` ` `{` ` ` `let bell = ` `new` `Array(n+1);` ` ` `for` `(let i = 0; i < n + 1; i++)` ` ` `{` ` ` `bell[i] = ` `new` `Array(n + 1);` ` ` `}` ` ` `bell[0][0] = 1;` ` ` ` ` `for` `(let 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` `(let j=1; j<=i; j++)` ` ` `bell[i][j] = bell[i-1][j-1] + bell[i][j-1];` ` ` `}` ` ` ` ` `return` `bell[n][0];` ` ` `}` ` ` ` ` `for` `(let n=0; n<=5; n++)` ` ` `document.write(` `"Bell Number "` `+ n + ` `" is "` `+bellNumber(n) + ` `"</br>"` `);` ` ` `</script>` |

**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(n^{2}). 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

This article is contributed by **Rajeev Agrawal**. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

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