论文标题
戴克路径中的对称峰和对称山谷
Symmetric peaks and symmetric valleys in Dyck paths
论文作者
论文摘要
Flórez和Rodr \'ıguez引入了Dyck路径中对称和不对称峰的概念，后者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功能来概括它们的结果，以跟踪对称峰的数量和不对称峰的数量以及这些峰的宽度。我们将丹妮丝和模拟的公式作为我们结果的特殊情况。 我们还考虑了对称山谷的类似但更复杂的概念。我们发现，相对于对称山谷的数量及其宽度的总和，dyck路径的生成函数的持续分数表达式，这在对称山谷与有序的根树有序的统计数据之间提供了意外的连接。最后，我们列举了其峰值或山谷高度满足某些单调性和非模式条件的堤防路径，并使用一个共同的框架恢复了一些已知结果，并将我们的问题与某些类别的柱状凸多个聚球体的列举联系起来。
The notion of symmetric and asymmetric peaks in Dyck paths was introduced by Flórez and Rodr\'ıguez, 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.
