Permutations with restricted movement

A restricted permutation of a locally finite directed graph G = (V, E) is a vertex permutation π : V → V for which (v, π(v)) ∈ E, for any vertex v ∈ V . The set of such permutations, denoted by ?(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 Z d permutations, in which ?(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 Z d-permutations. We discuss the global and local admissibility of patterns, in the context of restricted Z d-permutations. Finally, we review the related models of injective and surjective restricted functions.