论文标题
步行在方格上遵守两步规则：全，半和四分之一的飞机
Walks obeying two-step rules on the square lattice: full, half and quarter planes
论文作者
论文摘要
我们考虑步行在正方形晶格$ \ mathbb z^2 $的边缘上，遵守\ emph {两步规则，}，允许（或禁止）在给定方向上沿另一个方向的步骤。我们根据许多标准对这些规则进行分类，并显示这些属性如何影响其生成功能，渐近枚举和限制形状，以及上半平面。 对于四分之一飞机的散步，我们只迈出了一些初步的第一步。我们提出了模型组的候选者，类似于常规短步平面模型的组，并研究哪些模型具有有限的无限组。我们证明，用于求解许多原始模型的轨道总和可以在此处为某些模型工作，从而产生D-Finite解决方案。我们还为所有模型生成了短系列，并在可能的情况下猜测差异或代数方程。在这样做的过程中，我们发现这里有一些可能的短步骤模型的可能性，包括具有代数或d-finite生成函数但无限组的病例，以及具有非D-FINITE生成函数但有限组的模型。
We consider walks on the edges of the square lattice $\mathbb Z^2$ which obey \emph{two-step rules,} which allow (or forbid) steps in a given direction to be followed by steps in another direction. We classify these rules according to a number of criteria, and show how these properties affect their generating functions, asymptotic enumerations and limiting shapes, on the full lattice as well as the upper half plane. For walks in the quarter plane, we only make a few tentative first steps. We propose candidates for the group of a model, analogous to the group of a regular short-step quarter plane model, and investigate which models have finite versus infinite groups. We demonstrate that the orbit sum method used to solve a number of the original models can be made to work for some models here, producing a D-finite solution. We also generate short series for all models and guess differential or algebraic equations where possible. In doing so, we find that there are possibilities here which do not occur for the regular short-step models, including cases with algebraic or D-finite generating functions but infinite groups, as well as models with non-D-finite generating functions but finite groups.
