Permutations with restricted movement

TL;DR

This paper studies restricted permutations on graphs, especially on $bZ^d$, establishing a link with perfect matchings to analyze topological entropy and pattern admissibility in these dynamical systems.

Contribution

It introduces a correspondence between restricted permutations and perfect matchings, enabling entropy computation and pattern analysis in $bZ^d$-permutations.

Findings

Established a link between restricted permutations and perfect matchings.

Computed topological entropy for certain restricted $bZ^d$-permutations.

Analyzed global and local pattern admissibility in the context of these permutations.

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 (2016) 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.