629. K Inverse Pairs Array

📋 Đề Bài

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

🧠 Thuật Toán & Kỹ Thuật

Dynamic Programming (Quy hoạch động)Bit Manipulation (Thao tác bit)Matrix (Ma trận)

⏱️ Thời gian O(n×m)

💾 Không gian O(n×m)

💻 Lời Giải

C++ 0629-k-inverse-pairs-array.cpp

class Solution {
public:
    int kInversePairs(int n, int k) {
        const int mod = 1e9 + 7;
        vector<vector<int>> dp(n + 1, vector<int>(k + 1, 0));

        dp[0][0] = 1;

        for (int i = 1; i <= n; i++) {
            for (int j = 0; j <= k; j++) {
                for (int inv = 0; inv <= min(j, i - 1); inv++) {
                    dp[i][j] = (dp[i][j] + dp[i - 1][j - inv]) % mod;
                }
            }
        }

        return dp[n][k];
    }
};