Catalan numbers and their interpretations in terms of Dyck paths are widely used in different topics of applied mathematics and computer science. Here, we consider a general approach for constrained Dyck paths. In particular, we study Dyck paths of height at most $h$ with the additional restriction of having no $k-1$ consecutive valleys at height $h-1$. We give a combinatorial description of this class of paths and derive enumeration formulas using classical techniques for counting constrained lattice paths. As a consequence of this analysis, we obtain an identity involving Catalan numbers which, to the best of the authors' knowledge, does not appear in the existing literature. This identity arises naturally from the combinatorial interpretation and provides a new relation among families of Dyck paths with height and local structural constraints.

翻译：Catalan数及其通过Dyck路径的解释广泛应用于应用数学和计算机科学的各领域。本文考虑受限Dyck路径的一般方法。特别地，我们研究高度至多为$h$且附加限制条件为在高度$h-1$处无$k-1$个连续谷的Dyck路径。我们给出此类路径的组合描述，并利用经典计数受限格点路径的方法推导枚举公式。作为该分析的结果，我们得到一个涉及Catalan数的恒等式，据作者所知，该恒等式在现有文献中尚未出现。此恒等式自然源于组合解释，并揭示了具有高度约束和局部结构约束的Dyck路径族之间的新关系。