Permutations with restricted movement

Abstract

A restricted permutation of a locally finite directed graph $ G = (V, E) $ is a vertex permutation $ \pi: V\to V $ for which $ (v, \pi(v))\in E $, for any vertex $ v\in V $. The set of such permutations, denoted by $ \Omega(G) $, with a group action induced from a subset of graph isomorphisms form a topological dynamical system. We focus on the particular case presented by Schmidt and Strasser [18] of restricted $ {\mathbb Z}^d $ permutations, in which $ \Omega(G) $ is a subshift of finite type. We show a correspondence between restricted permutations and perfect matchings (also known as dimer coverings). We use this correspondence in order to investigate and compute the topological entropy in a class of cases of restricted $ {\mathbb Z}^d $-permutations. We discuss the global and local admissibility of patterns, in the context of restricted $ {\mathbb Z}^d $-permutations. Finally, we review the related models of injective and surjective restricted functions.