论文标题
晶格路径的对称程度
The degree of symmetry of lattice paths
论文作者
论文摘要
组合对象的对称程度（例如晶格路径）是对象的对称程度的度量。如果对象完全不对称，则通常从零开始，如果完全不对称，则它是完全对称的。我们研究了这种统计数据对戴克路径和大戴克路径的行为，并通过沿垂直线的反射描述了对称性。分区，与结合给出的对称性；某些构图被解释为bargraphs。我们找到了这些对象的对称程度以及其半长度或半级别计的生成函数的表达式，在大多数情况下，它们在渐近上可以推论出对称程度，对称程度具有瑞利（Rayleigh）或半符号的限制分布。所得的生成函数通常是代数，但明显的染料路径例外，我们根据我们使用的功能方程来猜测它是D-finite（但不是代数），我们使用的是使用Bijections在平面中行走的函数。
The degree of symmetry of a combinatorial object, such as a lattice path, is a measure of how symmetric the object is. It typically ranges from zero, if the object is completely asymmetric, to its size, if it is completely symmetric. We study the behavior of this statistic on Dyck paths and grand Dyck paths, with symmetry described by reflection along a vertical line through their midpoint; partitions, with symmetry given by conjugation; and certain compositions interpreted as bargraphs. We find expressions for the generating functions for these objects with respect to their degree of symmetry, and their semilength or semiperimeter, deducing in most cases that, asymptotically, the degree of symmetry has a Rayleigh or half-normal limiting distribution. The resulting generating functions are often algebraic, with the notable exception of Dyck paths, for which we conjecture that it is D-finite (but not algebraic), based on a functional equation that we obtain using bijections to walks in the plane.
