Permutations in two dimensions that maximally separate neighbors

Abstract

We characterize all permutations on even-by-even grids that maximally separate neighboring vertices. More precisely, let $n_1$, $n_2$ be positive even integers, let $I(n_1,n_2)=\{1,\dots,n_1\}\times\{1,\dots,n_2\}$ be the $n_1\times n_2$ grid, let $d$ be the $L_1$ metric on $I(n_1,n_2)$, and let $N=\{\{x,y\}\in I(n_1,n_2)\times I(n_1,n_2):d(x,y)=1\}$ be the set of neighbors in $I(n_1,n_2)$. We characterize all permutations $\pi$ of $I(n_1,n_2)$ that maximize $\sum_{\{x,y\}\in N} d(\pi(x),\pi(y))$.