标题： 在环面上排列分离相邻元素和矩形包装 显示英文标题

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

摘要： 设$n$、$s$和$k$为正整数。 对于不同的$i,j\in\mathbb{Z}_n$，定义$||i,j||_n$为当$\mathbb{Z}_n$的元素写成一个圆圈时，$i$和$j$之间的距离。 因此，\[ ||i,j||_n=\min\{(i-j)\bmod n,(j-i)\bmod n\}. \]一个排列$\pi:\mathbb{Z}_n\rightarrow\mathbb {Z}_n$是\emph{$(s,k)$-无冲突}如果$||\pi(i),\pi(j)||_n\geq k$当时$||i,j||_n<s$。 因此，一个$(s,k)$-无冲突的排列会将每一对靠近的元素（距离小于$s$）移动到一对距离较远的元素（距离至少为$k$）。 “$(s,k)$-无冲突的排列”的概念可以重新表述为在$n\times n$拓扑上某种$s\times k$矩形的排列。 对于整数$n$和$k$且$1\leq k<n$，令$\sigma(n,k)$为$s$的最大值，使得存在一个$(s,k)$-无冲突的排列$\mathbb{Z}_n$。 加强Blackburn最近的一篇论文，该论文证明了Mammoliti和Simpson的猜想，我们确定了$\sigma(n,k)$在所有情况下的值。 显示英文摘要

摘要： 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 \emph{$(s,k)$-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.