3577 - Count the Number of Computer Unlocking Permutations.

Medium

You are given an array complexity of length n.

There are n locked computers in a room with labels from 0 to n - 1, each with its own unique password. The password of the computer i has a complexity complexity[i].

The password for the computer labeled 0 is already decrypted and serves as the root. All other computers must be unlocked using it or another previously unlocked computer, following this information:

• You can decrypt the password for the computer i using the password for computer j, where j is any integer less than i with a lower complexity. (i.e. j < i and complexity[j] < complexity[i])

• To decrypt the password for computer i, you must have already unlocked a computer j such that j < i and complexity[j] < complexity[i].

Find the number of permutations of [0, 1, 2, ..., (n - 1)] that represent a valid order in which the computers can be unlocked, starting from computer 0 as the only initially unlocked one.

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

Note that the password for the computer with label 0 is decrypted, and not the computer with the first position in the permutation.

Example 1:

Input: complexity = 1,2,3

Output: 2

Explanation:

The valid permutations are:

• 0, 1, 2

• 0, 2, 1

Example 2:

Input: complexity = 3,3,3,4,4,4

Output: 0

Explanation:

There are no possible permutations which can unlock all computers.

Constraints:

• <code>2 <= complexity.length <= 10⁵</code>

• <code>1 <= complexityi<= 10⁹</code>