629. K Inverse Pairs Array

Problem Link

Description

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For an integer array nums, an inverse pair is a pair of integers [i, j] where 0 <= i < j < nums.length and nums[i] > nums[j].

Given two integers n and k, return the number of different arrays consisting of numbers from 1 to n such that there are exactly k inverse pairs. Since the answer can be huge, return it modulo 109 + 7.

 

Example 1:

Input: n = 3, k = 0
Output: 1
Explanation: Only the array [1,2,3] which consists of numbers from 1 to 3 has exactly 0 inverse pairs.

Example 2:

Input: n = 3, k = 1
Output: 2
Explanation: The array [1,3,2] and [2,1,3] have exactly 1 inverse pair.

 

Constraints:

• 1 <= n <= 1000
• 0 <= k <= 1000

Solution

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Python3

class Solution:
    def kInversePairs(self, n: int, k: int) -> int:
        
        max_possible_inversions = (n * (n-1)//2)
        if k > max_possible_inversions:
            return 0
        if k == 0 or k == max_possible_inversions:
            return 1
        
        MOD = 10 ** 9 + 7
        
        dp = [[0]*(k+1) for _ in range(n+1)]
        
        for i in range(1, n+1):
            dp[i][0] = 1
 
        dp[2][1] = 1
        
        for i in range(3,n+1):
            max_possible_inversions = min(k, i*(i-1)//2)
            for j in range(1,  max_possible_inversions + 1):
                dp[i][j] = dp[i][j-1] + dp[i-1][j] 
                if j>=i:
                    dp[i][j] -= dp[i-1][j - i]
                dp[i][j] = (dp[i][j] + MOD) % MOD
            
        return dp[n][k]