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.

中文翻译：

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运动受限的排列

局部有限有向图$ G =（V，E）$的受限置换是一个顶点置换$ \ pi：V \至V $，其中$（v，\ pi（v））\在E $中，对于任何顶点$ v \ in V $。此类置换的集合（用$ \ Omega（G）$表示）具有从图同构子集诱发的组动作，从而形成拓扑动力学系统。我们专注于Schmidt和Strasser提出的特殊案例[18岁]的受限$ {\\ mathbb Z} ^ d $排列，其中$ \ Omega（G）$是有限类型的子移位。我们显示了受限排列和完美匹配（也称为二聚体覆盖物）之间的对应关系。我们使用此对应关系来调查和计算在受限$ {\ mathbb Z} ^ d $置换情况下的一类情况下的拓扑熵。我们在受限制的$ {\ mathbb Z} ^ d $置换的背景下讨论模式的全局和局部可采性。最后，我们回顾了内射和外射受限功能的相关模型。