Permutations that Separate Close Elements, and Rectangle Packings in the Torus

Let $n$, $s$ and $k$ be positive integers. For distinct $i,j\in\mathbb{Z}_n$, define $||i,j||_n$ to be the distance between $i$ and $j$ when the elements of $\mathbb{Z}_n$ are written in a circle. So\[||i,j||_n=\min\{(i-j)\bmod n,(j-i)\bmod n\}.\]A permutation $\pi:\mathbb{Z}_n\rightarrow\mathbb {Z}_n$ is $(s,k)$-clash-free if $||\pi(i),\pi(j)||_n\geq k$ whenever $||i,j||_n