Count the number of ways to divide N in k groups incrementally

Difficulty Level : Hard
Last Updated : 26 Oct, 2021

Given two integers N and K, the task is to count the number of ways to divide N into K groups of positive integers such that their sum is N and the number of elements in groups follows a non-decreasing order (i.e group[i] <= group[i+1]).
Examples:

Input: N = 8, K = 4
Output: 5
Explanation:
Their are 5 groups such that their sum is 8 and the number of positive integers in each group is 4.
[1, 1, 1, 5], [1, 1, 2, 4], [1, 1, 3, 3], [1, 2, 2, 3], [2, 2, 2, 2]
Input: N = 24, K = 5
Output: 164
Explanation:
There are 164 such groups such that their sum is 24 and number of positive integers in each group is 5

Different Approaches:
1. Naive Approach (Time: O(N^K), Space: O(N))
2. Memoization (Time: O((N³*K), Space: O(N²*K))
3. Bottom-Up Dynamic Programming (Time: O(N*K), Space: O(N*K))

Naive Approach: We can solve this problem using recursion. At each step of recursion put all the values greater than equal to the previously computed value.
Below is the implementation of the above approach:

C++

// C++ implementation to count the
// number of ways to divide N in
// groups such that each group
// has K number of elements
 
#include <bits/stdc++.h>
 
using namespace std;
 
// Function to count the number
// of ways to divide the number N
// in groups such that each group
// has K number of elements
int calculate(int pos, int prev,
                 int left, int k)
{
    // Base Case
    if (pos == k) {
        if (left == 0)
            return 1;
        else
            return 0;
    }
 
    // if N is divides completely
    // into less than k groups
    if (left == 0)
        return 0;
 
    int answer = 0;
     
    // put all possible values
    // greater equal to prev
    for (int i = prev; i <= left; i++) {
        answer += calculate(pos + 1, i,
                          left - i, k);
    }
    return answer;
}
 
// Function to count the number of
// ways to divide the number N
int countWaystoDivide(int n, int k)
{
    return calculate(0, 1, n, k);
}
 
// Driver Code
int main()
{
    int N = 8;
    int K = 4;
 
    cout << countWaystoDivide(N, K);
    return 0;
}

Java

// Java implementation to count the
// number of ways to divide N in
// groups such that each group
// has K number of elements
import java.util.*;
class GFG{
 
// Function to count the number
// of ways to divide the number N
// in groups such that each group
// has K number of elements
static int calculate(int pos, int prev,
                     int left, int k)
{
     
    // Base Case
    if (pos == k)
    {
        if (left == 0)
            return 1;
        else
            return 0;
    }
 
    // If N is divides completely
    // into less than k groups
    if (left == 0)
        return 0;
 
    int answer = 0;
     
    // Put all possible values
    // greater equal to prev
    for(int i = prev; i <= left; i++)
    {
       answer += calculate(pos + 1, i,
                           left - i, k);
    }
    return answer;
}
 
// Function to count the number of
// ways to divide the number N
static int countWaystoDivide(int n, int k)
{
    return calculate(0, 1, n, k);
}
 
// Driver Code
public static void main(String[] args)
{
    int N = 8;
    int K = 4;
 
    System.out.print(countWaystoDivide(N, K));
}
}
 
// This code is contributed by Rajput-Ji

Python3

# Python3 implementation to count the
# number of ways to divide N in
# groups such that each group
# has K number of elements
 
# Function to count the number
# of ways to divide the number N
# in groups such that each group
# has K number of elements
def calculate(pos, prev, left, k):
     
    # Base Case
    if (pos == k):
        if (left == 0):
            return 1;
        else:
            return 0;
 
    # If N is divides completely
    # into less than k groups
    if (left == 0):
        return 0;
 
    answer = 0;
 
    # Put all possible values
    # greater equal to prev
    for i in range(prev, left + 1):
        answer += calculate(pos + 1, i,
                           left - i, k);
 
    return answer;
 
# Function to count the number of
# ways to divide the number N
def countWaystoDivide(n, k):
     
    return calculate(0, 1, n, k);
 
# Driver Code
if __name__ == '__main__':
     
    N = 8;
    K = 4;
 
    print(countWaystoDivide(N, K));
 
# This code is contributed by 29AjayKumar

C#

// C# implementation to count the
// number of ways to divide N in
// groups such that each group
// has K number of elements
using System;
 
class GFG{
 
// Function to count the number
// of ways to divide the number N
// in groups such that each group
// has K number of elements
static int calculate(int pos, int prev,
                     int left, int k)
{
     
    // Base Case
    if (pos == k)
    {
        if (left == 0)
            return 1;
        else
            return 0;
    }
 
    // If N is divides completely
    // into less than k groups
    if (left == 0)
        return 0;
 
    int answer = 0;
     
    // Put all possible values
    // greater equal to prev
    for(int i = prev; i <= left; i++)
    {
       answer += calculate(pos + 1, i,
                           left - i, k);
    }
    return answer;
}
 
// Function to count the number of
// ways to divide the number N
static int countWaystoDivide(int n, int k)
{
    return calculate(0, 1, n, k);
}
 
// Driver Code
public static void Main(String[] args)
{
    int N = 8;
    int K = 4;
 
    Console.Write(countWaystoDivide(N, K));
}
}
 
// This code is contributed by Rajput-Ji

Javascript

<script>
 
// Javascript implementation to count the
// number of ways to divide N in
// groups such that each group
// has K number of elements
 
 
 
// Function to count the number
// of ways to divide the number N
// in groups such that each group
// has K number of elements
function calculate(pos, prev,
                left, k)
{
    // Base Case
    if (pos == k) {
        if (left == 0)
            return 1;
        else
            return 0;
    }
 
    // if N is divides completely
    // into less than k groups
    if (left == 0)
        return 0;
 
    let answer = 0;
     
    // put all possible values
    // greater equal to prev
    for (let i = prev; i <= left; i++) {
        answer += calculate(pos + 1, i,
                        left - i, k);
    }
    return answer;
}
 
// Function to count the number of
// ways to divide the number N
function countWaystoDivide(n, k)
{
    return calculate(0, 1, n, k);
}
 
// Driver Code
 
    let N = 8;
    let K = 4;
 
    document.write(countWaystoDivide(N, K));
     
// This code is contributed by Mayank Tyagi
     
</script>

Output

Time complexity: O(N^K)
Auxiliary Space: O(N).
Memoization Approach: In the previous approach we can see that we are solving the subproblems repeatedly, i.e. it follows the property of Overlapping Subproblems. So we can memoize the same using the DP table.
Below is the implementation of the above approach:

C++

// C++ implementation to count the
// number of ways to divide N in
// groups such that each group
// has K number of elements
 
#include <bits/stdc++.h>
 
using namespace std;
 
// DP Table
int dp[500][500][500];
 
// Function to count the number
// of ways to divide the number N
// in groups such that each group
// has K number of elements
int calculate(int pos, int prev,
                int left, int k)
{
    // Base Case
    if (pos == k) {
        if (left == 0)
            return 1;
        else
            return 0;
    }
    // if N is divides completely
    // into less than k groups
    if (left == 0)
        return 0;
 
    // If the subproblem has been
    // solved, use the value
    if (dp[pos][prev][left] != -1)
        return dp[pos][prev][left];
 
    int answer = 0;
    // put all possible values
    // greater equal to prev
    for (int i = prev; i <= left; i++) {
        answer += calculate(pos + 1, i,
                           left - i, k);
    }
 
    return dp[pos][prev][left] = answer;
}
 
// Function to count the number of
// ways to divide the number N in groups
int countWaystoDivide(int n, int k)
{
    // Initialize DP Table as -1
    memset(dp, -1, sizeof(dp));
 
    return calculate(0, 1, n, k);
}
 
// Driver Code
int main()
{
    int N = 8;
    int K = 4;
 
    cout << countWaystoDivide(N, K);
    return 0;
}

Java

// Java implementation to count the
// number of ways to divide N in
// groups such that each group
// has K number of elements
import java.util.*;
class GFG{
  
// DP Table
static int [][][]dp = new int[500][500][500];
  
// Function to count the number
// of ways to divide the number N
// in groups such that each group
// has K number of elements
static int calculate(int pos, int prev,
                     int left, int k)
{
    // Base Case
    if (pos == k)
    {
        if (left == 0)
            return 1;
        else
            return 0;
    }
   
    // if N is divides completely
    // into less than k groups
    if (left == 0)
        return 0;
  
    // If the subproblem has been
    // solved, use the value
    if (dp[pos][prev][left] != -1)
        return dp[pos][prev][left];
  
    int answer = 0;
   
    // put all possible values
    // greater equal to prev
    for (int i = prev; i <= left; i++)
    {
        answer += calculate(pos + 1, i,
                           left - i, k);
    }
  
    return dp[pos][prev][left] = answer;
}
  
// Function to count the number of
// ways to divide the number N in groups
static int countWaystoDivide(int n, int k)
{
    // Initialize DP Table as -1
        for (int i = 0; i < 500; i++)
        {
            for (int j = 0; j < 500; j++)
            {
                for (int l = 0; l < 500; l++)
                    dp[i][j][l] = -1;
            }
        }
  
    return calculate(0, 1, n, k);
}
  
// Driver Code
public static void main(String[] args)
{
    int N = 8;
    int K = 4;
  
    System.out.print(countWaystoDivide(N, K));
}
}
 
// This code is contributed by Rajput-Ji

Python3

# Python3 implementation to count the
# number of ways to divide N in
# groups such that each group
# has K number of elements
  
# DP Table
dp = [[[0 for i in range(50)]
             for j in range(50)]
          for j in range(50)]
 
# Function to count the number
# of ways to divide the number N
# in groups such that each group
# has K number of elements
def calculate(pos, prev, left, k):
    
    # Base Case
    if (pos == k):
        if (left == 0):
            return 1;
        else:
            return 0;
  
    # if N is divides completely
    # into less than k groups
    if (left == 0):
        return 0;
  
    # If the subproblem has been
    # solved, use the value
    if (dp[pos][prev][left] != -1):
        return dp[pos][prev][left];
  
    answer = 0;
  
    # put all possible values
    # greater equal to prev
    for i in range(prev,left+1):
        answer += calculate(pos + 1, i,
                            left - i, k);
    dp[pos][prev][left] = answer;
    return dp[pos][prev][left];
  
# Function to count the number of
# ways to divide the number N in groups
def countWaystoDivide(n, k):
   
    # Initialize DP Table as -1
    for i in range(50):
        for j in range(50):
            for l in range(50):
                dp[i][j][l] = -1;
  
    return calculate(0, 1, n, k);
  
# Driver Code
if __name__ == '__main__':
    N = 8;
    K = 4;
  
    print(countWaystoDivide(N, K));
  
# This code is contributed by Rajput-Ji

C#

// C# implementation to count the
// number of ways to divide N in
// groups such that each group
// has K number of elements
using System;
class GFG{
  
// DP Table
static int [,,]dp = new int[50, 50, 50];
  
// Function to count the number
// of ways to divide the number N
// in groups such that each group
// has K number of elements
static int calculate(int pos, int prev,
                     int left, int k)
{
    // Base Case
    if (pos == k)
    {
        if (left == 0)
            return 1;
        else
            return 0;
    }
   
    // if N is divides completely
    // into less than k groups
    if (left == 0)
        return 0;
  
    // If the subproblem has been
    // solved, use the value
    if (dp[pos, prev, left] != -1)
        return dp[pos, prev, left];
  
    int answer = 0;
   
    // put all possible values
    // greater equal to prev
    for (int i = prev; i <= left; i++)
    {
        answer += calculate(pos + 1, i,
                           left - i, k);
    }
  
    return dp[pos, prev, left] = answer;
}
  
// Function to count the number of
// ways to divide the number N in groups
static int countWaystoDivide(int n, int k)
{
    // Initialize DP Table as -1
        for (int i = 0; i < 50; i++)
        {
            for (int j = 0; j < 50; j++)
            {
                for (int l = 0; l < 50; l++)
                    dp[i, j, l] = -1;
            }
        }
  
    return calculate(0, 1, n, k);
}
  
// Driver Code
public static void Main(String[] args)
{
    int N = 8;
    int K = 4;
  
    Console.Write(countWaystoDivide(N, K));
}
}
 
// This code is contributed by gauravrajput1

Javascript

<script>
 
// JavaScript implementation to count the
// number of ways to divide N in
// groups such that each group
// has K number of elements
 
// DP Table
let dp = new Array(500);
for(let i=0;i<500;i++)
{
    dp[i]=new Array(500);
    for(let j=0;j<500;j++)
    {
        dp[i][j]=new Array(500);
        for(let k=0;k<500;k++)
            dp[i][j][k]=0;
    }
}
 
// Function to count the number
// of ways to divide the number N
// in groups such that each group
// has K number of elements
function calculate(pos,prev,left,k)
{
    // Base Case
    if (pos == k)
    {
        if (left == 0)
            return 1;
        else
            return 0;
    }
    
    // if N is divides completely
    // into less than k groups
    if (left == 0)
        return 0;
   
    // If the subproblem has been
    // solved, use the value
    if (dp[pos][prev][left] != -1)
        return dp[pos][prev][left];
   
    let answer = 0;
    
    // put all possible values
    // greater equal to prev
    for (let i = prev; i <= left; i++)
    {
        answer += calculate(pos + 1, i,
                           left - i, k);
    }
   
    return dp[pos][prev][left] = answer;
}
 
// Function to count the number of
// ways to divide the number N in groups
function countWaystoDivide(n,k)
{
    // Initialize DP Table as -1
        for (let i = 0; i < 500; i++)
        {
            for (let j = 0; j < 500; j++)
            {
                for (let l = 0; l < 500; l++)
                    dp[i][j][l] = -1;
            }
        }
   
    return calculate(0, 1, n, k);
}
 
// Driver Code
let N = 8;
let K = 4;
 
document.write(countWaystoDivide(N, K));
 
 
// This code is contributed by unknown2108
 
</script>

Output:

Time complexity: O(N^3 * K)
Auxiliary Space: O(N^2 * K).

Bottom-Up DP: We are asked to find CountWaystoDivide(n,k) So the recurrence approach and explanation of DP is:

CountWaystoDivide( n , k ) = CountWaystoDivide( n-k , k ) + CountWaystoDivide( n-1 , k-1 )
Explanation:
Divide CountWaystoDivide( n , k ) into two parts where
If first element is 1 then the rest form a total of n-1 divide into k-1 so CountWaystoDivide( n-1 , k-1 )
If first element is greater than 1 then, we can subtract 1 from every element and get a valid partition of n-k into k parts, hence CountWaystoDivide( n-1 , k-1 ).
Mathematical Explanation of DP:
As each group must have at least one person, so, give each group one person, then we are left with n-k persons, which can we divided into 1,2,3..or k groups. Thus our dp will be: dp[n][k] = dp[n-k][1] + dp[n-k][2] + dp[n-k][3] + …. + dp[n-k][k].
At first look, the previous might give O(N³) vibes, but with a little manipulation we can optimize it:
dp[n][k] = dp[(n-1)-(k-1)][1] + dp[(n-1)-(k-1)][2] + … + dp[(n-1)-(k-1)][k-1] + dp[(n-1)-(k-1)][k]
From the recurrence, we can write:
dp[n][k] = dp[n-1][k-1] + dp[n-k][k]

Steps to solve the problem using DP:

Initialize a 2-D array dp[][] of size n+1, k+1 where dp[i][j] will store the optimal solution to divide n into k groups.
For each value from i=0 to n, dp[n][1] will be 1 since total ways to divide n into 1 is 1. also dp[0][0] will be 1.

DP states are updated as follows:

If i>=j then dp[i][j] = dp[i-1][j-1] + dp[i-j][j]
otherwise i-j<0 and dp[i-j][j] become zero so dp[i][j] = dp[i-1][j-1]

C++

// C++ implementation to count the
// number of ways to divide N in
// groups such that each group
// has K number of elements
 
#include <bits/stdc++.h>
 
using namespace std;
 
// Function to count the number of
// ways to divide the number N in groups
int countWaystoDivide(int n, int k)
{
    if (n < k)
        return 0; // When n is less than k, No way to divide
                  // into groups
    vector<vector<int> > dp(n + 1, vector<int>(k + 1));
 
    for (int i = 1; i <= n; i++)
        dp[i][1]
            = 1; // exact one way to divide n to 1 group
    dp[0][0] = 1;
 
    for (int i = 1; i <= n; i++) {
        for (int j = 2; j <= k; j++) {
            if (i >= j)
                dp[i][j] = dp[i - j][j] + dp[i - 1][j - 1];
            else
                dp[i][j]
                    = dp[i - 1][j - 1]; // i<j so dp[i-j][j]
                                        // becomes zero
        }
    }
    return dp[n][k]; // returning number of ways to divide N
                     // in k groups
}
 
// Driver Code
int main()
{
    int N = 8;
    int K = 4;
 
    cout << countWaystoDivide(N, K);
    return 0;
}

Java

// Java implementation to count the
// number of ways to divide N in
// groups such that each group
// has K number of elements
import java.io.*;
 
class GFG {
 
  static int countWaystoDivide(int n, int k)
  {
    if (n < k)
      return 0; // When n is less than k, No way to divide
    // into groups
 
    int [][]dp = new int[n+1][k+1];
    for (int i = 1; i <= n; i++)
      dp[i][1]
      = 1; // exact one way to divide n to 1 group
    dp[0][0] = 1;
 
    for (int i = 1; i <= n; i++) {
      for (int j = 2; j <= k; j++) {
        if (i >= j)
          dp[i][j] = dp[i - j][j] + dp[i - 1][j - 1];
        else
          dp[i][j]
          = dp[i - 1][j - 1]; // i<j so dp[i-j][j]
        // becomes zero
      }
    }
    return dp[n][k]; // returning number of ways to divide N
    // in k groups
  }
 
  // Driver code
  public static void main (String[] args)
  {
    int N = 8;
    int K = 4;
 
    System.out.println(countWaystoDivide(N, K));
  }
}
 
// This code is contributed by rohitsingh07052.

Python3

# Python3 implementation to count the
# number of ways to divide N in
# groups such that each group
# has K number of elements
 
# DP Table
# Function to count the number of
# ways to divide the number N in groups
 
 
def countWaystoDivide(n, k):
    if(n < k):
        return 0
 
    dp = [[0 for i in range(k+1)] for i in range(n+1)]
    for i in range(1, n+1):
        dp[i][1] = 1
    dp[0][0] = 1
    for i in range(1, n+1):
        for j in range(2, k+1):
            if(i >= j):
                dp[i][j] = dp[i-1][j-1] + dp[i-j][j]
 
            else:
                dp[i][j] = dp[i-1][j-1]
    return dp[n][k]
 
 
# Driver Code
if __name__ == '__main__':
    N = 8
    K = 4
 
    print(countWaystoDivide(N, K))
 
# This code is contributed by Rajput-Ji

Javascript

<script>
    // Javascript implementation to count the
    // number of ways to divide N in
    // groups such that each group
    // has K number of elements
     
    // Function to count the number of
    // ways to divide the number N in groups
    function countWaystoDivide(n, k)
    {
        if (n < k)
            return 0; // When n is less than k, No way to divide
                      // into groups
        let dp = new Array(n + 1);
        for(let i = 0; i < n + 1; i++)
        {
            dp[i] = new Array(k + 1);
            for(let j = 0; j < k + 1; j++)
            {
                dp[i][j] = 0;
            }
        }
 
        for (let i = 1; i <= n; i++)
            dp[i][1]
                = 1; // exact one way to divide n to 1 group
        dp[0][0] = 1;
 
        for (let i = 1; i <= n; i++) {
            for (let j = 2; j <= k; j++) {
                if (i >= j)
                    dp[i][j] = dp[i - j][j] + dp[i - 1][j - 1];
                else
                    dp[i][j]
                        = dp[i - 1][j - 1]; // i<j so dp[i-j][j]
                                            // becomes zero
            }
        }
        return dp[n][k]; // returning number of ways to divide N
                         // in k groups
    }
     
    let N = 8;
    let K = 4;
  
    document.write(countWaystoDivide(N, K));
 
// This code is contributed by mukesh07.
</script>

Output

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

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