We initiate the study of the enumerative combinatorics of the intervals in the Dyck pattern poset. More specifically, we find some closed formulas to express the size of some specific intervals, as well as the number of their covering relations. In most of the cases, we are also able to refine our formulas by rank. We also provide the first results on the Möbius function of the Dyck pattern poset, giving for instance a closed expression for the Möbius function of initial intervals whose maximum is a Dyck path having exactly two peaks.

中文翻译：

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戴克模式集的区间的枚举组合

我们开始研究Dyck模式波普中的区间的枚举组合。更具体地说，我们找到了一些封闭的公式来表示某些特定区间的大小以及它们的覆盖关系的数量。在大多数情况下，我们还可以按等级细化公式。我们还提供了有关Dyck模式摆球的Möbius函数的第一个结果，例如，给出了初始间隔的Möbius函数的封闭表达式，其初始间隔的最大值是具有两个正好两个峰值的Dyck路径。