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Eulerian Number in C++
In mathematics, the Eulerian number is a special type of combination number. It defines the number of permutations in which the next element is a specific number greater than the previous one.
Denoted as,
A(n, m) is permutation from 1 to n in which two numbers vary by m.
Problem Statement: In this problem, we are given two numbers m and n. And we need to find the number of permutations that are the Eulerian Number.
Let’s take an example to understand the problem,
Input: n = 4, m = 2
Output: 11
Explanation:
All permutations of number from 1 to 4 are −
1 2 3 4 1 2 4 3 1 3 2 4 1 3 4 2 1 4 2 3 1 4 3 2
2 1 3 4 2 1 4 3 2 3 1 4 2 3 4 1 2 4 1 3 2 4 3 1
3 1 2 4 3 1 4 2 3 2 1 4 3 2 4 1 3 4 1 2 3 4 2 1
4 1 2 3 4 1 3 2 4 2 1 3 4 2 3 1 4 3 1 2 4 3 2 1
Out of all 11 permutations have the difference between two numbers m.
Solution Approach −
To find the number of permutations, we will be using the formula for eulerian number,
A(n, m) = 0, if m > n or n = 0
A(n, m) = 1, if m = 0
A(n, m) = (n-m)A(n-1, m-1) + (m+1)A(n-1, m)
Program to illustrate the working of our solution,
Example
#include <iostream>
using namespace std;
int countEulerianNumber(int n, int m)
{
if (m >= n || n == 0)
return 0;
if (m == 0)
return 1;
return ( ( (n - m) * countEulerianNumber(n - 1, m - 1) ) + ( (m + 1) * countEulerianNumber(n - 1, m) ) );
}
int main() {
int n = 5, m = 3;
cout<<"The number of Eulerian permutations is "<<countEulerianNumber(n, m);
return 0;
}Output −
The number of Eulerian permutations is 26
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