Number of permutation with K inversions | Set 2

Given two integers N and K, the task is to count the number of permutations of the first N natural numbers having exactly K inversions. Since the count can be very large, print it modulo 10⁹ + 7.

An inversion is defined as a pair a[i], a[j] such that a[i] > a[j] and i < j.

Examples:

Input: N = 3, K = 2
Output: 2
Explanation:
All Permutations for N = 3 are 321, 231, 213, 312, 132, 123.
Out of which only 231 and 312 have 2 inversions as:

• 231: 2 > 1 & 3 > 1
• 312: 3 > 1 & 3 > 2.

Therefore, both are satisfying the condition of having exactly K inversions.

Input: N = 5, K = 5
Output: 22

Naive Approach: Refer to the previous post for the simplest possible approach to solve the problem. 
Time Complexity: O(N*N!)
Auxiliary Space: O(1)

Dynamic Programming using Top-Down Approach: Refer to the previous post of this article for the Top-Down Approach. 
Time Complexity: O(N*K²)
Auxiliary Space: O(N*K)

Dynamic Programming using Bottom-Up Approach:

Illustration:

For Example: N = 4, K = 2

N - 1 = 3, K₀ = 0 ...   123           =>  1423
N - 1 = 3, K₁ = 1 ...   213, 132   =>  2143, 1342
N - 1 = 3, K₂ = 2 ...   231, 312   =>  2314, 3124
So the answer is 5.

The maximum value is taken between (K - N + 1) and 0 as K inversions cannot be obtained if the number of inversions in permutation of (N - 1) numbers is less than K - (N - 1) as at most (N - 1) new inversions can be obtained by adding N^th number at the beginning.

Follow the steps below to solve the problem:

• Create an auxiliary array dp[2][K + 1] where dp[N][K] stores all permutations of (N - 1) numbers with K = (max(K - (N - 1), 0) to K) inversions, by adding N^th number with them only once.
• Using dp[i % 2][K] will interchange iteration between two rows and take j = Max(K - (N - 1), 0). So dp[N[K] = dp[N-1][j] + dp[N-1][j+1] + .... + dp[N - 1][K].
• For calculating dp[N][K] there is no need to do this extra K iteration as it can be obtained in O(1) from dp[N][K - 1]. So the recurrence relation is given by:
  • dp[N][K] = dp[N][K - 1] + dp[N - 1][K] - dp[N - 1][max(K - (N - 1), 0) - 1]
• Iterate two nested loops using the variable i and j over N and K respectively and update each dp states as per the above recurrence relation.
• Print the value of dp[N%2][K] after the above steps as the result.

Below is the implementation of the above approach:

// C++ program for the above approach

#include <bits/stdc++.h>
using namespace std;

// Function to count permutations with
// K inversions
int numberOfPermWithKInversion(
    int N, int K)
{
    // Store number of permutations
    // with K inversions
    int dp[2][K + 1];

    int mod = 1000000007;

    for (int i = 1; i <= N; i++) {
        for (int j = 0; j <= K; j++) {

            // If N = 1 only 1 permutation
            // with no inversion
            if (i == 1)
                dp[i % 2][j] = (j == 0);

            // For K = 0 only 1 permutation
            // with no inversion
            else if (j == 0)
                dp[i % 2][j] = 1;

            // Otherwise Update each dp
            // state as per the reccurrance
            // relation formed
            else
                dp[i % 2][j]
                    = (dp[i % 2][j - 1] % mod
                       + (dp[1 - i % 2][j]
                          - ((max(j - (i - 1), 0) == 0)
                                 ? 0
                                 : dp[1 - i % 2]
                                     [max(j - (i - 1), 0)
                                      - 1])
                          + mod)
                             % mod)
                      % mod;
            ;
        }
    }

    // Print final count
    cout << dp[N % 2][K];
}

// Driver Code
int main()
{
    // Given N and K
    int N = 3, K = 2;

    // Function Call
    numberOfPermWithKInversion(N, K);

    return 0;
}

// Java program for the above approach
import java.io.*;

class GFG{

// Function to count permutations with
// K inversions
static void numberOfPermWithKInversion(int N, int K)
{
    
    // Store number of permutations
    // with K inversions
    int[][] dp = new int[2][K + 1];

    int mod = 1000000007;

    for(int i = 1; i <= N; i++) 
    {
        for(int j = 0; j <= K; j++) 
        {
            
            // If N = 1 only 1 permutation
            // with no inversion
            if (i == 1)
            {
                dp[i % 2][j] = (j == 0) ? 1 : 0;
            }

            // For K = 0 only 1 permutation
            // with no inversion
            else if (j == 0)
                dp[i % 2][j] = 1;

            // Otherwise Update each dp
            // state as per the reccurrance
            // relation formed
            else
                {
                 int maxm = Math.max(j - (i - 1));
             dp[i % 2][j] = (dp[i % 2][j - 1] % mod +
                               (dp[1 - i % 2][j] - 
                               ((Math.max(j - (i - 1), 0) == 0) ?
                                   0 : dp[1 - i % 2][maxm, 0) - 1]) +
                                         mod) % mod) % mod;
}
        }
    }
    
    // Print final count
    System.out.println (dp[N % 2][K]);
}

// Driver Code
public static void main(String[] args)
{
    
    // Given N and K
    int N = 3, K = 2;

    // Function Call
    numberOfPermWithKInversion(N, K);
}
}

// This code is contributed by akhilsaini

# Python3 program for the above approach

# Function to count permutations with
# K inversions
def numberOfPermWithKInversion(N, K):
    
    # Store number of permutations
    # with K inversions
    dp = [[0] * (K + 1)] * 2

    mod = 1000000007

    for i in range(1, N + 1):
        for j in range(0, K + 1):
            
            # If N = 1 only 1 permutation
            # with no inversion
            if (i == 1):
                dp[i % 2][j] = 1 if (j == 0) else 0

            # For K = 0 only 1 permutation
            # with no inversion
            elif (j == 0):
                dp[i % 2][j] = 1

            # Otherwise Update each dp
            # state as per the reccurrance
            # relation formed
            else:
                var = (0 if (max(j - (i - 1), 0) == 0)
                         else dp[1 - i % 2][max(j - (i - 1), 0) - 1])
                dp[i % 2][j] = ((dp[i % 2][j - 1] % mod +
                                (dp[1 - i % 2][j] -
                                (var) + mod) % mod) % mod)

    # Print final count
    print(dp[N % 2][K])

# Driver Code
if __name__ == '__main__':
    
    # Given N and K
    N = 3
    K = 2

    # Function Call
    numberOfPermWithKInversion(N, K)

# This code is contributed by akhilsaini

// C# program for the above approach
using System;

class GFG{

// Function to count permutations with
// K inversions
static void numberOfPermWithKInversion(int N, int K)
{
    
    // Store number of permutations
    // with K inversions
    int[,] dp = new int[2, K + 1];

    int mod = 1000000007;

    for(int i = 1; i <= N; i++)
    {
        for(int j = 0; j <= K; j++) 
        {
            
            // If N = 1 only 1 permutation
            // with no inversion
            if (i == 1)
            {
                dp[i % 2, j] = (j == 0) ? 1 : 0;
            }

            // For K = 0 only 1 permutation
            // with no inversion
            else if (j == 0)
                dp[i % 2, j] = 1;

            // Otherwise Update each dp
            // state as per the reccurrance
            // relation formed
            else
                dp[i % 2, j] = (dp[i % 2, j - 1] % mod + 
                               (dp[1 - i % 2, j] - 
                               ((Math.Max(j - (i - 1), 0) == 0) ?
                                   0 : dp[1 - i % 2, Math.Max(
                                          j - (i - 1), 0) - 1]) +
                                              mod) % mod) % mod;
        }
    }

    // Print final count
    Console.WriteLine(dp[N % 2, K]);
}

// Driver Code
public static void Main()
{

    // Given N and K
    int N = 3, K = 2;

    // Function Call
    numberOfPermWithKInversion(N, K);
}
}

// This code is contributed by akhilsaini

<script>

// Javascript program to implement
// the above approach

// Function to count permutations with
// K inversions
function numberOfPermWithKInversion(N, K)
{
      
    // Store number of permutations
    // with K inversions
    let dp = new Array(2);
    // Loop to create 2D array using 1D array
    for (var i = 0; i < dp.length; i++) 
    {
    dp[i] = new Array(2);
    }
  
    let mod = 1000000007;
  
    for(let i = 1; i <= N; i++) 
    {
        for(let j = 0; j <= K; j++) 
        {
              
            // If N = 1 only 1 permutation
            // with no inversion
            if (i == 1)
            {
                dp[i % 2][j] = (j == 0) ? 1 : 0;
            }
  
            // For K = 0 only 1 permutation
            // with no inversion
            else if (j == 0)
                dp[i % 2][j] = 1;
  
            // Otherwise Update each dp
            // state as per the reccurrance
            // relation formed
            else
                dp[i % 2][j] = (dp[i % 2][j - 1] % mod +
                 (dp[1 - i % 2][j] - 
                 ((Math.max(j - (i - 1), 0) == 0) ?
                   0 : dp[1 - i % 2][(Math.max(j - 
                          (i - 1), 0) - 1)]) +
                             mod) % mod) % mod;
        }
    }
      
    // Print final count
    document.write(dp[N % 2][K]);
}
    
    // Driver Code
     
           // Given N and K
    let N = 3, K = 2;
  
    // Function Call
    numberOfPermWithKInversion(N, K);
  
</script>

2

Time Complexity: O(N * K)
Auxiliary Space: O(K)