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Let $\mathcal{T}^{(p)}_n$ be the set of $p$-ary labeled trees on$\{1,2,\dots,n\}$. A maximal decreasing subtree of an $p$-ary labeled tree isdefined by the maximal $p$-ary subtree from the root with all edges beingdecreasing. In this paper, we study a new refinement $\mathcal{T}^{(p)}_{n,k}$of $\mathcal{T}^{(p)}_n$, which is the set of $p$-ary labeled trees whosemaximal decreasing subtree has $k$ vertices.

A refined enumeration of $p$-ary labeled trees