${\mathcal{D}}_n$ is a finite set

Given: a natural number $n$.

${\mathcal{D}}_n$ is the set of diagonal paths of length $2n$ from $(0,0)$ to $(2n,0)$ whose height function satisfies $h(i)\ge 0$ for every $i$ with $0\le i\le 2n$ (Dyck paths of semilength $n$).

A diagonal path of length $\mathrm{\ell }$ from $(0,\alpha )$ is the same datum as a function $h:\{0,\dots ,\mathrm{\ell }\}\to \mathbb{Z}$ with $h(0)=\alpha$ and $h(i)-h(i-1)\in \{1,-1\}$ for $1\le i\le \mathrm{\ell }$ (Diagonal lattice paths with steps $U=(1,1)$ and $D=(1,-1)$, and the height function).

For a step set $S$, a point $P$ and $\mathrm{\ell }\in \mathbb{N}$, the map sending a lattice path to its step word is a bijection ${\mathcal{L}}_S(P;\mathrm{\ell })\to S^{\mathrm{\ell }}$, and ${\mathcal{L}}_S(P;\mathrm{\ell })$ is finite with $\mid {\mathcal{L}}_S(P;\mathrm{\ell })\mid =\mid S{\mid }^{\mathrm{\ell }}$ (For each start point the step word is a bijection onto $S^n$).

For finite sets $A$ and $B$, the set $A^B$ of functions $B\to A$ is finite and $\mid A^B\mid =\mid A{\mid }^{\mid B\mid }$ (The set $A^B$ of functions $B\to A$ between finite sets is finite, with $\mid A^B\mid =\mid A{\mid }^{\mid B\mid }$).

A subset of a finite set is finite, and a subset $S$ of a finite set $A$ has $\mid S\mid \le \mid A\mid$ (A subset of a finite set is finite, with $\mid B\mid \le \mid A\mid$, and equality holds if and only if $B=A$, clauses 1 and 2).

A property that holds at $0$ and passes from every natural number to its successor holds at every natural number: if a property $P$ satisfies $P(0)$ and ($P(n)\Rightarrow P(\sigma (n))$) for all $n$, then $P(n)$ holds for all $n\in \mathbb{N}$ (The principle of mathematical induction).