3470 - Permutations IV.

Hard

Given two integers, n and k, an alternating permutation is a permutation of the first n positive integers such that no two adjacent elements are both odd or both even.

Return the k-th alternating permutation sorted in lexicographical order. If there are fewer than k valid alternating permutations , return an empty list.

Example 1:

Input: n = 4, k = 6

Output: 3,4,1,2

Explanation:

The lexicographically-sorted alternating permutations of [1, 2, 3, 4] are:

• [1, 2, 3, 4]

• [1, 4, 3, 2]

• [2, 1, 4, 3]

• [2, 3, 4, 1]

• [3, 2, 1, 4]

• [3, 4, 1, 2] ← 6th permutation

• [4, 1, 2, 3]

• [4, 3, 2, 1]

Since k = 6, we return [3, 4, 1, 2].

Example 2:

Input: n = 3, k = 2

Output: 3,2,1

Explanation:

The lexicographically-sorted alternating permutations of [1, 2, 3] are:

• [1, 2, 3]

• [3, 2, 1] ← 2nd permutation

Since k = 2, we return [3, 2, 1].

Example 3:

Input: n = 2, k = 3

Output: []

Explanation:

The lexicographically-sorted alternating permutations of [1, 2] are:

• [1, 2]

• [2, 1]

There are only 2 alternating permutations, but k = 3, which is out of range. Thus, we return an empty list [].

Constraints:

• 1 <= n <= 100

• <code>1 <= k <= 10¹⁵</code>