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 $\mathrm{\mid }\mathrm{\mid }i,j\mathrm{\mid }{\mathrm{\mid }}_n$ to be the distance between $i$ and $j$ when the elements of ${\mathbb{Z}}_n$ are written in a circle. So $$\mathrm{\mid }\mathrm{\mid }i,j\mathrm{\mid }{\mathrm{\mid }}_n=\min \{(i-j)n,(j-i)n\}\mathrm{.}$$ A permutation $\pi :{\mathbb{Z}}_n\to {\mathbb{Z}}_n$ is \emph{$(s,k)$-clash-free} if $\mathrm{\mid }\mathrm{\mid }\pi (i),\pi (j)\mathrm{\mid }{\mathrm{\mid }}_n\ge k$ whenever $\mathrm{\mid }\mathrm{\mid }i,j\mathrm{\mid }{\mathrm{\mid }}_n<s$. So an $(s,k)$-clash-free permutation moves every pair of close elements (at distance less than $s$) to a pair of elements at large distance (at distance at least $k$). The notion of an $(s,k)$-clash-free permutation can be reformulated in terms of certain packings of $s\times k$ rectangles on an $n\times n$ torus. For integers $n$ and $k$ with $1\le k<n$, let $\sigma (n,k)$ be the largest value of $s$ such that an $(s,k)$-clash-free permutation of ${\mathbb{Z}}_n$ exists. Strengthening a recent paper of Blackburn, which proved a conjecture of Mammoliti and Simpson, we determine the value of $\sigma (n,k)$ in all cases.