count-the-number-of-computer-unlocking-permutations

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]
  
  
  
  • Unlock computer 0 first with root password.
  
  
  • Unlock computer 1 with password of computer 0 since complexity[0] < complexity[1].
  
  
  • Unlock computer 2 with password of computer 1 since complexity[1] < complexity[2].
  
  
  
  


• [0, 2, 1]
  
  
  
  • Unlock computer 0 first with root password.
  
  
  • Unlock computer 2 with password of computer 0 since complexity[0] < complexity[2].
  
  
  • Unlock computer 1 with password of computer 0 since complexity[0] < complexity[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:

• 2 <= complexity.length <= 105


• 1 <= complexity[i] <= 109