Beautiful Arrangement (M)

题目

Suppose you have n integers from 1 to n. We define a beautiful arrangement as an array that is constructed by these n numbers successfully if one of the following is true for the ith position (1 <= i <= n) in this array:

• The number at the ith position is divisible by i.
• i is divisible by the number at the ith position.

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]:
Number at the 1st position (i=1) is 1, and 1 is divisible by i (i=1).
Number at the 2nd position (i=2) is 2, and 2 is divisible by i (i=2).
The second beautiful arrangement is [2, 1]:
Number at the 1st position (i=1) is 2, and 2 is divisible by i (i=1).
Number at the 2nd position (i=2) is 1, and i (i=2) is divisible by 1.

Example 2:

Input: n = 1
Output: 1

Constraints:

• 1 <= n <= 15

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题意

对1-n这n个数进行排列，使得对于序列中第i个数字x满足i是x的倍数或者x是i的倍数。

思路

回溯法，对1-n每个位置挑选一个满足的数字放上去，判断最终得到的序列是否有效。

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代码实现

Java

class Solution {
    public int countArrangement(int n) {
        return dfs(1, n, new boolean[n + 1]);
    }

    private int dfs(int index, int n, boolean[] used) {
        if (index == n + 1) {
            return 1;
        }

        int count = 0;
        for (int i = 1; i <= n; i++) {
            if (!used[i] && (index % i == 0 || i % index == 0)) {
                used[i] = true;
                count += dfs(index + 1, n, used);
                used[i] = false;
            }
        }

        return count;
    }
}