the following property $$\label{admissible} \text{$P$ has a peak $z$ at height $b$} \Rightarrow \text{$\text{ht}_{z'}(P)>a$ for all $z'\in \text{supp}(P)$ with $\ell_{z'}(P)\geq \ell_{z}(P)$}.$$ With other words, once the path has a peak at height $b$, the remaining subsequent path is strictly above the line $y=a$ (see Figure \[fig55\] for a pictorial illustration).

(0,0) grid (16,6); (0,0) – (3,3); (3,3) – (4,4) – (5,3); (4,1) – (16,1); at (0,1) [$y=a$]{}; at (0,4) [$y=b$]{};

 {#section-4}

For $n,m\in {\mathbb Z}_+$ let $n_0,n_1\in{\mathbb Z}_+$ such that $n_0<m$ and $n=m n_1+n_0$. *In what follows we set $N=\max\{m-1,n\}$.* We are interested in a suitable subset of $\mathcal{D}^N_n$ which we will call admissible Dyck path. If $n<m$ we set $A(m,n)=\emptyset$ and otherwise define $$A(m,n):=\{(i_1,m),(i_2,m+1),\dots,(i_{n-m+1},n)\}\subset {\mathbb Z}_+^2$$ where $i_1<\cdots<i_{n-m+1}$ is the natural ordering of the set $$\{0,\dots,n\}\backslash \{pn_1+n_0+\min\{0,(p-1)-n_0\},\ 1\leq p\leq m\}.$$ Equivalently, $$\begin{aligned}
A(m,n)=\notag &\{(p(n_1+1)+r,m+pn_1+r),\ 0\leq r<n_1,\ 0\leq p\leq n_0\}\ \cup &\\& \notag \{(pn_1+(n_0+1)+r,m+p(n_1-1)+(n_0+1)+r),\ 0\leq r<n_1-1,\ n_0< p<m\}.\end{aligned}$$ The following is straightforward: $$\label{1}
(a,b)\in A(m,n)\backslash\{(0,m)\} \Rightarrow \exists \ \tilde a<a: \ \  (\tilde a,b-1)\in A(m,n).$$

\[maindef\] We call a path $P\in \mathcal{D}_n^N$ admissible iff $P$ is $(a,b)$–admissible for all $(a,b)\in A(m,n)$. We denote by $\mathcal{D}_{m,n}$ the set of admissible Dyck paths and set $\mathcal{D}_{m,n}(s):=\mathcal{D}_n^N(s)\cap \mathcal{D}_{m,n}$.

Both sets will play a 