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

Simon R. Blackburn ; Tuvi Etzion¹

¹Technion -- Israel Institute of Technology

The electronic journal of combinatorics, Tome 31 (2024) no. 4

Résumé

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 $||i,j||_n<s$. 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 $1\leq k<n$, 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.

Détail

DOI : 10.37236/12711

Classification : 05B40, 05A05
Mots-clés : \((s, k)\)-clash-free permutation, cyclic matching sequencibility