The notion of symmetric and asymmetric peaks in Dyck paths was introduced by Flórez and Rodríguez, who counted the total number of such peaks over all Dyck paths of a given length. In this paper we generalize their results by giving multivariate generating functions that keep track of the number of symmetric peaks and the number of asymmetric peaks, as well as the widths of these peaks. We recover a formula of Denise and Simion as a special case of our results.

We also consider the analogous but more intricate notion of symmetric valleys. We find a continued fraction expression for the generating function of Dyck paths with respect to the number of symmetric valleys and the sum of their widths, which provides an unexpected connection between symmetric valleys and statistics on ordered rooted trees. Finally, we enumerate Dyck paths whose peak or valley heights satisfy certain monotonicity and unimodality conditions, using a common framework to recover some known results, and relating our questions to the enumeration of certain classes of column-convex polyominoes.

Flórez和Rodríguez引入了Dyck路径中对称峰和非对称峰的概念，他们计算了给定长度的所有Dyck路径上此类峰的总数。在本文中，我们通过提供多元生成函数来概括其结果，这些函数可以跟踪对称峰的数量和非对称峰的数量以及这些峰的宽度。我们恢复了Denise和Simion的公式，作为结果的特例。

我们还考虑对称谷的类似但更复杂的概念。对于对称谷的数量及其宽度之和，我们发现了Dyck路径的生成函数的连续分数表达式，这为对称谷与有序根树的统计数据之间提供了意料之外的联系。最后，我们使用共同的框架来恢复某些已知结果，并将其峰高或谷高满足某些单调性和单峰性条件的Dyck路径枚举，并将我们的问题与某些类型的列凸多胺基体的枚举相关联。