A refinement for ordered labeled trees

Let ${\mathcal{O}}_n$ be the set of ordered labeled trees on $0,\mathrm{.}\mathrm{.}\mathrm{.},n$. A maximal decreasing subtree of an ordered labeled tree is defined by the maximal ordered subtree from the root with all edges being decreasing. In this paper, we study a new refinement ${\mathcal{O}}_{n,k}$ of ${\mathcal{O}}_n$, which is the set of ordered labeled trees whose maximal decreasing subtree has $k+1$ vertices.