On the poset of King-Non-Attacking permutations

A king-non-attacking permutation is a permutation $\pi \in S_n$ such that $\mathrm{\mid }\pi (i)-\pi (i-1)\mathrm{\mid }\mathrm{\ne }1$ for each $i\in \{2,\dots ,n\}$. We investigate the structure of the poset of these permutations under the containment relation, and also provide some results on its M"obius function.