Permutations that separate close elements, and rectangle packings in the torus

Abstract

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 $\pi :{\mathbb{Z}}_n\to {\mathbb{Z}}_n$ is \emph{$(s,k)$-clash-free} if $||\pi (i),\pi (j)|{|}_n\ge k$ whenever $||i,j|{|}_n<s$. 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.

21 pages, 6 figures