论文标题
从二聚体模型到广义晶格路径
From dimer models to generalized lattice paths
论文作者
论文摘要
斐波那契类型的二聚体模型的生成函数的复发关系提供了与与晶格路径相关的正式功率序列的功能关系，例如Dyck，Motzkin和Schröder路径。在本文中，我们通过修改二聚体模型的复发关系，将其概括为通用晶格路径，$ k $ -dyck，$ k $ -motzkin和$ k $-schröder路径的信件。我们通过保持其组合结构来介绍五种二聚体模型的概括。这使我们能够根据广义晶格路径表达生成功能。给定的晶格路径的重量涉及多个统计数据，例如大小，面积，峰和山谷以及水平步骤的高度。我们通过使用复发关系和拉格朗日定理来列举广义晶格路径。
A recurrence relation of the generating function of the dimer model of Fibonacci type gives a functional relation for formal power series associated to lattice paths such as a Dyck, Motzkin and Schröder path. In this paper, we generalize the correspondence to the case of generalized lattice paths, $k$-Dyck, $k$-Motzkin and $k$-Schröder paths, by modifying the recurrence relation of the dimer model. We introduce five types of generalizations of the dimer model by keeping its combinatorial structures. This allows us to express the generating functions in terms of generalized lattice paths. The weight given to a generalized lattice path involves several statistics such as size, area, peaks and valleys, and heights of horizontal steps. We enumerate the generalized lattice paths by use of the recurrence relations and the Lagrange inversion theorem.
