  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.
