Enumeration of chains and saturated chains in Dyck lattices

We find a closed formula for the number of chains in Dyck lattices. Moreover, we determine a general formula to compute the number of saturated chains, and we apply it to find the number of saturated chains of length 2, 3 and 4. We also compute what we call the Hasse index (of order 2, 3 and 4) of Dyck lattices, which is the ratio between the total number of saturated chains (of length 2, 3 and 4) and the cardinality of the underlying poset.