Given two integers n and k, return the $k^{ ext{th}}$ lexicographically sorted alternating permutation of the integers from $1$ to n. If there are fewer than k valid alternating permutations, return an empty list [].

A permutation of the integers from $1$ to n is considered alternating if the parity of adjacent elements alternates, meaning no two adjacent elements are both even or both odd.

Constraints:

• $1 \leq$ n $\leq 100$

• $1 \leq$ k $\leq 10^{15}$

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