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Eulerian Number

In combinatorics, the Eulerian Number A(n, m), is the number of permutations of the numbers 1 to n in which exactly m elements are greater than previous element.

For example, there are 4 permutations of the number 1 to 3 in which exactly 1 element is greater than the previous elements.
Eulerian

Examples :

Input : n = 3, m = 1
Output : 4
Please see above diagram (There
are 4 permutations where 1 no. is
greater.

Input : n = 4, m = 1
Output : 11



Eulerian Numbers are the coefficients of the Eulerian polynomials described below.

The Eulerian polynomials are defined by the exponential generating function

The Eulerian polynomials can be computed by the recurrence

An explicit formula for A(n, m) is

We can calculate A(n, m) by recurrence relation:

Example:
Suppose, n = 3 and m = 1.
Therefore,
A(3, 1)
= (3 – 1) * A(2, 0) + (1 + 1) * A(2, 1)
= 2 * A(2, 0) + 2 * A(2, 1)
= 2 * 1 + 2 * ( (2 – 1) * A(1, 0) + (1 + 1) * A(1, 1))
= 2 + 2 * (1 * 1 + 2 * ((1 – 1) * A(0, 0) + (1 + 1) * A(0, 1))
= 2 + 2 * (1 + 2 * (0 * 1 + 2 * 0)
= 2 + 2 * (1 + 2 * 0)
= 2 + 2 * 1
= 2 + 2
= 4
We can verify this with example shown above.

Below is the implementation of finding A(n, m):

C++

// CPP Program to find Eulerian number A(n, m)
#include <bits/stdc++.h>
using namespace std;
  
// Return euleriannumber A(n, m)
int eulerian(int n, int m)
{
    if (m >= n || n == 0)
        return 0;
  
    if (m == 0)
        return 1;
  
    return (n - m) * eulerian(n - 1, m - 1) + 
           (m + 1) * eulerian(n - 1, m);
}
  
// Driven Program
int main()
{
    int n = 3, m = 1;
    cout << eulerian(n, m) << endl;
    return 0;
}

Java

// Java rogram to find Eulerian number A(n, m)
import java.util.*;
  
class Eulerian
{
    // Return eulerian number A(n, m)
    public static int eulerian(int n, int m)
    {
        if (m >= n || n == 0)
            return 0;
  
        if (m == 0)
            return 1;
  
        return (n - m) * eulerian(n - 1, m - 1) +
            (m + 1) * eulerian(n - 1, m);
    }
      
    // driver code    
    public static void main(String[] args)
    {
        int n = 3, m = 1;
        System.out.print( eulerian(n, m) );
    }
}
  
// This code is contributed by rishabh_jain

Python3

# Python3 Program to find Eulerian number A(n, m)
  
# Return euleriannumber A(n, m)
def eulerian(n, m):
    if (m >= n or n == 0):
        return 0;
  
    if (m == 0):
        return 1;
  
    return ((n - m) * eulerian(n - 1, m - 1) +
            (m + 1) * eulerian(n - 1, m))
  
# Driver code
n = 3
m = 1
print( eulerian(n, m) )
  
# This code is contributed by rishabh_jain

C#

// C# rogram to find Eulerian number A(n, m)
using System;
  
class Eulerian {
      
    // Return eulerian number A(n, m)
    public static int eulerian(int n, int m)
    {
        if (m >= n || n == 0)
            return 0;
  
        if (m == 0)
            return 1;
  
        return (n - m) * eulerian(n - 1, m - 1) + 
                    (m + 1) * eulerian(n - 1, m);
    }
  
    // driver code
    public static void Main()
    {
        int n = 3, m = 1;
        Console.WriteLine(eulerian(n, m));
    }
}
  
// This code is contributed by vt_m

PHP

<?php
// PHP Program to find
// Eulerian number A(n, m)
  
// Return euleriannumber A(n, m)
function eulerian($n, $m)
{
    if ($m >= $n || $n == 0)
        return 0;
  
    if ($m == 0)
        return 1;
  
    return ($n - $m) * eulerian($n - 1, $m - 1) + 
                 ($m + 1) * eulerian($n - 1, $m);
}
  
// Driven Code
$n = 3; $m = 1;
echo eulerian($n, $m);
  
// This code is contributed by anuj_67.
?>


Output :

4

Below is the implementation of finding A(n, m) using Dynamic Programming:

C++

// CPP Program to find Eulerian number A(n, m)
#include <bits/stdc++.h>
using namespace std;
  
// Return euleriannumber A(n, m)
int eulerian(int n, int m)
{
    int dp[n + 1][m + 1];
  
    memset(dp, 0, sizeof(dp));
  
    // For each row from 1 to n
    for (int i = 1; i <= n; i++) {
  
        // For each column from 0 to m
        for (int j = 0; j <= m; j++) {
  
            // If i is greater than j
            if (i > j) {
  
                // If j is 0, then make that 
                // state as 1.
                if (j == 0)
                    dp[i][j] = 1;
  
                // basic recurrence relation.
                else
                    dp[i][j] = ((i - j) *
                     dp[i - 1][j - 1]) + 
                    ((j + 1) * dp[i - 1][j]);
            }
        }
    }
  
    return dp[n][m];
}
  
// Driven Program
int main()
{
    int n = 3, m = 1;
    cout << eulerian(n, m) << endl;
    return 0;
}

Java

// Java rogram to find Eulerian number A(n, m)
import java.util.*;
  
class Eulerian
{
    // Return euleriannumber A(n, m)
    public static int eulerian(int n, int m)
    {
        int[][] dp = new int[n+1][m+1];
  
        // For each row from 1 to n
        for (int i = 1; i <= n; i++) {
      
            // For each column from 0 to m
            for (int j = 0; j <= m; j++) {
  
                // If i is greater than j
                if (i > j) {
  
                    // If j is 0, then make 
                    // that state as 1.
                    if (j == 0)
                        dp[i][j] = 1;
  
                    // basic recurrence relation.
                    else
                        dp[i][j] = ((i - j) *
                            dp[i - 1][j - 1]) +
                        ((j + 1) * dp[i - 1][j]);
                }
            }
        }
  
        return dp[n][m];
    }
      
    // driver code
    public static void main(String[] args)
    {
        int n = 3, m = 1;
        System.out.print( eulerian(n, m) );
    }
}
  
// This code is contributed by rishabh_jain

Python3

# Python3 Program to find Eulerian 
# number A(n, m)
  
# Return euleriannumber A(n, m)
def eulerian(n, m):
    dp = [[0 for x in range(m+1)] 
             for y in range(n+1)] 
  
    # For each row from 1 to n
    for i in range(1, n+1):
  
        # For each column from 0 to m
        for j in range(0, m+1):
  
            # If i is greater than j
            if (i > j):
                # If j is 0, then make that 
                # state as 1.
  
                if (j == 0):
                    dp[i][j] = 1
  
                # basic recurrence relation.
                else :
                    dp[i][j] = (((i - j) * 
                       dp[i - 1][j - 1]) + 
                       ((j + 1) * dp[i - 1][j]))
                      
    return dp[n][m]
  
# Driven Program
n = 3
m = 1
print(eulerian(n, m))
  
# This code is contributed by Prasad Kshirsagar

C#

// C# rogram to find Eulerian number A(n, m)
using System;
  
class Eulerian {
      
    // Return euleriannumber A(n, m)
    public static int eulerian(int n, int m)
    {
        int[, ] dp = new int[n + 1, m + 1];
  
        // For each row from 1 to n
        for (int i = 1; i <= n; i++) {
  
            // For each column from 0 to m
            for (int j = 0; j <= m; j++) {
  
                // If i is greater than j
                if (i > j) {
  
                    // If j is 0, then make
                    // that state as 1.
                    if (j == 0)
                        dp[i, j] = 1;
  
                    // basic recurrence relation.
                    else
                        dp[i, j] = ((i - j) * dp[i - 1, j - 1]) +
                                        ((j + 1) * dp[i - 1, j]);
                }
            }
        }
  
        return dp[n, m];
    }
  
    // driver code
    public static void Main()
    {
        int n = 3, m = 1;
        Console.WriteLine(eulerian(n, m));
    }
}
  
// This code is contributed by vt_m

PHP

<?php
// PHP Program to find Eulerian
// number A(n, m) 
  
// Return euleriannumber A(n, m) 
function eulerian($n, $m
    $dp = array(array());
      
    for ($i = 0; $i < $n + 1; $i++)
        for($j = 0; $j < $m + 1; $j++)
            $dp[$i][$j] = 0 ;
  
    // For each row from 1 to n 
    for ($i = 1; $i <= $n; $i++) 
    
  
        // For each column from 0 to m 
        for ($j = 0; $j <= $m; $j++) 
        
  
            // If i is greater than j 
            if ($i > $j
            
  
                // If j is 0, then make that 
                // state as 1. 
                if ($j == 0) 
                    $dp[$i][$j] = 1; 
  
                // basic recurrence relation. 
                else
                    $dp[$i][$j] = (($i - $j) * 
                                    $dp[$i - 1][$j - 1]) + 
                                  (($j + 1) * $dp[$i - 1][$j]); 
            
        
    
  
    return $dp[$n][$m]; 
  
// Driver Code
$n = 3 ;
$m = 1; 
echo eulerian($n, $m) ; 
  
// This code is contributed by Ryuga
?>


Output :

4


This article is attributed to GeeksforGeeks.org

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