  It is proved that for every positive integer $n$, the number of
non-Tukey-equivalent directed sets of cardinality $\leq \aleph_n$ is at least
$c_{n+2}$, the $(n+2)$-Catalan number. Moreover, the Tukey class $\mathcal
D_{\aleph_n} $ of directed sets of cardinality $\leq \aleph_n$ contains an
isomorphic copy of the poset of Dyck $(n+2)$-paths. Furthermore, we give a
complete description whether two successive elements in the copy contain
another directed set in between or not.
