3193 - Count the Number of Inversions.

Hard

You are given an integer n and a 2D array requirements, where <code>requirementsi = end_i, cnt_i</code> 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], <code>perm0..end_i</code> has exactly <code>cnt_i</code> inversions.

Since the answer may be very large, return it modulo <code>10⁹ + 7</code>.

Example 1:

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

Output: 2

Explanation:

The two permutations are:

• [2, 0, 1]

• [1, 2, 0]

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

• <code>requirementsi = end_i, cnt_i</code>

• <code>0 <= end_i<= n - 1</code>

• <code>0 <= cnt_i<= 400</code>

• The input is generated such that there is at least one i such that <code>end_i == n - 1</code>.

• The input is generated such that all <code>end_i</code> are unique.