Abstract A Dyck path is a lattice path in the first quadrant of the x y -plane that starts at the origin and ends on the x -axis and has even length. This is composed of the same number of North-East ( X ) and South-East ( Y ) steps. A peak and a valley of a Dyck path are the subpaths X Y and Y X , respectively. A peak is symmetric if the valleys determining the maximal pyramid containing the peak are at the same level. In this paper we give recursive relations, generating functions, as well as closed formulas to count the total number of symmetric peaks and asymmetric peaks. We also give an asymptotic expansion for the number of symmetric peaks.

中文翻译：

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枚举 Dyck 路径中的对称和非对称峰

摘要 戴克路径是在 xy 平面的第一象限中从原点开始到 x 轴结束并且长度为偶数的晶格路径。这由相同数量的东北 ( X ) 和东南 ( Y ) 台阶组成。Dyck 路径的峰和谷分别是子路径 XY 和 YX 。如果确定包含峰的最大金字塔的谷处于同一水平，则峰是对称的。在本文中，我们给出了递归关系、生成函数以及封闭公式来计算对称峰和非对称峰的总数。我们还给出了对称峰数量的渐近展开式。