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