3193. Count the Number of Inversions

Description

You are given an integer n and a 2D array requirements, where requirements[i] = [end_i, cnt_i] represents the end index and the inversion count of each requirement.

A pair of indices (i, j) from an integer array nums is called an inversion if:

i < j and nums[i] > nums[j]

Return the number of permutations perm of [0, 1, 2, ..., n - 1] such that for all requirements[i], perm[0..end_i] has exactly cnt_i inversions.

Since the answer may be very large, return it modulo 10⁹ + 7.

Example 1:

Input: n = 3, requirements = [[2,2],[0,0]]

Output: 2

Explanation:

The two permutations are:

[2, 0, 1]
- Prefix [2, 0, 1] has inversions (0, 1) and (0, 2).
- Prefix [2] has 0 inversions.
[1, 2, 0]
- Prefix [1, 2, 0] has inversions (0, 2) and (1, 2).
- Prefix [1] has 0 inversions.

Example 2:

Input: n = 3, requirements = [[2,2],[1,1],[0,0]]

Output: 1

Explanation:

The only satisfying permutation is [2, 0, 1]:

Prefix [2, 0, 1] has inversions (0, 1) and (0, 2).
Prefix [2, 0] has an inversion (0, 1).
Prefix [2] has 0 inversions.

Example 3:

Input: n = 2, requirements = [[0,0],[1,0]]

Output: 1

Explanation:

The only satisfying permutation is [0, 1]:

Prefix [0] has 0 inversions.
Prefix [0, 1] has an inversion (0, 1).

Constraints:

2 <= n <= 300
1 <= requirements.length <= n
requirements[i] = [end_i, cnt_i]
0 <= end_i <= n - 1
0 <= cnt_i <= 400
The input is generated such that there is at least one i such that end_i == n - 1.
The input is generated such that all end_i are unique.

Solutions

Solution 1: Dynamic Programming

We define $f[i][j]$ as the number of permutations of $[0..i]$ with $j$ inversions. Consider the relationship between the number $a_i$ at index $i$ and the previous $i$ numbers. If $a_i$ is smaller than $k$ of the previous numbers, then each of these $k$ numbers forms an inversion pair with $a_i$, contributing to $k$ inversions. Therefore, we can derive the state transition equation:

$$ f[i][j] = \sum_{k=0}^{\min(i, j)} f[i-1][j-k] $$

Since the problem requires the number of inversions in $[0..\textit{end}_i]$ to be $\textit{cnt}_i$, when we calculate for $i = \textit{end}_i$, we only need to compute $f[i][\textit{cnt}_i]$. The rest of $f[i][..]$ will be $0$.

The time complexity is $O(n \times m \times \min(n, m))$, and the space complexity is $O(n \times m)$. Here, $m$ is the maximum number of inversions, and in this problem, $m \le 400$.

Python3JavaC++GoTypeScript

class Solution:
    def numberOfPermutations(self, n: int, requirements: List[List[int]]) -> int:
        req = [-1] * n
        for end, cnt in requirements:
            req[end] = cnt
        if req[0] > 0:
            return 0
        req[0] = 0
        mod = 10**9 + 7
        m = max(req)
        f = [[0] * (m + 1) for _ in range(n)]
        f[0][0] = 1
        for i in range(1, n):
            l, r = 0, m
            if req[i] >= 0:
                l = r = req[i]
            for j in range(l, r + 1):
                for k in range(min(i, j) + 1):
                    f[i][j] = (f[i][j] + f[i - 1][j - k]) % mod
        return f[n - 1][req[n - 1]]

class Solution {
    public int numberOfPermutations(int n, int[][] requirements) {
        int[] req = new int[n];
        Arrays.fill(req, -1);
        int m = 0;
        for (var r : requirements) {
            req[r[0]] = r[1];
            m = Math.max(m, r[1]);
        }
        if (req[0] > 0) {
            return 0;
        }
        req[0] = 0;
        final int mod = (int) 1e9 + 7;
        int[][] f = new int[n][m + 1];
        f[0][0] = 1;
        for (int i = 1; i < n; ++i) {
            int l = 0, r = m;
            if (req[i] >= 0) {
                l = r = req[i];
            }
            for (int j = l; j <= r; ++j) {
                for (int k = 0; k <= Math.min(i, j); ++k) {
                    f[i][j] = (f[i][j] + f[i - 1][j - k]) % mod;
                }
            }
        }
        return f[n - 1][req[n - 1]];
    }
}