beautiful_arrangement.py

"""
Suppose you have n integers labeled 1 through n. A permutation of those n integers perm (1-indexed) is considered a beautiful arrangement if for every i (1 <= i <= n), either of the following is true:

perm[i] is divisible by i.
i is divisible by perm[i].
Given an integer n, return the number of the beautiful arrangements that you can construct.

 

Example 1:

Input: n = 2
Output: 2
Explanation: 
The first beautiful arrangement is [1,2]:
    - perm[1] = 1 is divisible by i = 1
    - perm[2] = 2 is divisible by i = 2
The second beautiful arrangement is [2,1]:
    - perm[1] = 2 is divisible by i = 1
    - i = 2 is divisible by perm[2] = 1
Example 2:

Input: n = 1
Output: 1
 

Constraints:

1 <= n <= 15
"""

class Solution:
    def countArrangement(self, n: int) -> int:
        def dfs(index, sol):
            nonlocal res
            if len(sol) == n:
                res = res + 1
                return
            
            for num in range(1, n + 1):
                if num not in sol and (num % index == 0 or index % num == 0):
                    sol.append(num)
                    dfs(index + 1, sol)
                    sol.pop()
        res = 0
        sol = []
        dfs(1, [])
        return res