2 problems

Let $n,k,r\in\mathbb{N}$ with $k<n$, and let $\mathsf{LD}(n)^{\bullet k}$ be the partially labelled Dyck paths with $k$ decorated valleys. For $P$ in this set, let…

Let $m,n,k\in\mathbb{N}$ with $k<n$, and let $\mathsf{LD}(m,n)^{\bullet k}$ denote the partially labelled rectangular Dyck paths with $k$ decorated valleys. For such a path $P$, le…