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Bell Numbers - Number of ways to Partition a Set in C++
A bell number is used to denote the number of ways a set of n elements can be partitioned into subsets that are not empty (i.e. have at least one element).
In this program, we are given a set of n elements and we have to find the number of ways to partition the set into non-empty subsets.
Example
Input : 3 Output : 5
Explanation − let the set of three elements {1, 2, 3}.
The subsets are {{1} , {2} , {3}} ; {{1} , {2,3}} ; {{1 , 2} , {3}} ; {{2} , {1 , 3}} ; {1 , 2 , 3}.
Bell number: A bell number bell(n) gives the value of sum of s(n,k) for all values of k ranging from 1 to n. Here s(n,k) is the number of partitions of n elements into k subsets.
The formula would be −
$$bell(n)=\sum_{k=0}^n S(n,k)$$
The function s(n,k) is recursively as −
s(n+1,k) = k*s(n,k) + s(n,k-1).
Working
On adding (n+1)th element to k partitions, there are two possibilities −
It adds one to the k partitions of existing partitions i.e. s(n,k-1).
Adding value to all sets of partition, i.e. k*s(n,k).
First few Bell numbers are 1 , 1 , 2 ,5 ,15 , 52 , 205
For finding the Bells number, we have a few methods,
Simple method is to one by one compute s(n,k) for k = 1 to n and add sum of all values of the number.
Using bell triangle is to use bell’s triangle as given below −
1 1 2 2 3 5 5 7 10 15 15 20 27 37 52
Example
#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++) {
bell[i][0] = bell[i-1][i-1];
for (int j=1; j<=i; j++)
bell[i][j] = bell[i-1][j-1] + bell[i][j-1];
}
return bell[n][0];
}
int main() {
for (int n=0; n<=5; n++)
cout<<"Bell Number "<<n<<" is "<< bellNumber(n)<<endl;
return 0;
}
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