Enumerative combinatorics of intervals in the Dyck pattern poset

TL;DR

This paper explores the combinatorial structure of intervals within the Dyck pattern poset, providing formulas for their sizes, covering relations, and initial M"obius function values, advancing understanding of Dyck path poset properties.

Contribution

It introduces the first formulas for interval sizes, covering relations, and M"obius function values in the Dyck pattern poset, with refinements by rank.

Findings

Closed formulas for interval sizes

Formulas for covering relations

M"obius function for initial intervals with two peaks

Abstract

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 M\"obius function of the Dyck pattern poset, giving for instance a closed expression for the M\"obius function of initial intervals whose maximum is a Dyck path having exactly two peaks.