526. Beautiful Arrangement

lru_cache + dfs + bit mask

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

分析：

dfs + lru_cache+ bit mask

class Solution:
    def countArrangement(self, n: int) -> int:
        @lru_cache(maxsize=None)  
        def dfs(pos,mask):
            if pos == n+1:
                return 1
            res = 0
            for j in range(1,n+1):
                if not mask & (1<<j) and (j%i==0 or i%j==0):
                    res += dfs(i+1, mask | (1<<j))
            return res
        
        return dfs(1,0)