Enumeration of saturated chains in Dyck lattices

TL;DR

This paper presents a general formula for counting saturated chains in Dyck lattices, specifically calculating their numbers for lengths 2 and 3, and introduces the Hasse index for these lattices.

Contribution

It provides a novel general formula for saturated chains in Dyck lattices and computes specific counts and indices for lengths 2 and 3.

Findings

Derived a formula for saturated chains in Dyck lattices

Calculated the number of chains of length 2 and 3

Computed the Hasse index for these chains

Abstract

We determine a general formula to compute the number of saturated chains in Dyck lattices, and we apply it to find the number of saturated chains of length 2 and 3. We also compute what we call the Hasse index (of order 2 and 3) of Dyck lattices, which is the ratio between the total number of saturated chains (of length 2 and 3) and the cardinality of the underlying poset.