K Inverse Pairs Array — LeetCode 629 Python Solution

The problem

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.

Example

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.

Python solution

class Solution:
    def kInversePairs(self, n: int, k: int) -> int:
        mod = 10**9 + 7
        f = [1] + [0] * k
        s = [0] * (k + 2)
        for i in range(1, n + 1):
            for j in range(1, k + 1):
                f[j] = (s[j + 1] - s[max(0, j - (i - 1))]) % mod
            for j in range(1, k + 2):
                s[j] = (s[j - 1] + f[j - 1]) % mod
        return f[k]

Complexity

Pattern: Dynamic Programming

Define a state, write the transition, and stop recomputing the same subproblem. LeetCode 629. K Inverse Pairs Array is filed here on both counts: the reference solution below belongs to the algorithm family this hub collects, and LeetCode tags it Dynamic Programming.

The dynamic programming guide has the Python template for the pattern and the 481 LeetCode problems that use it.

Related problems