I've noticed that for every natural number $n\in\mathbb{N}$, there is a finite Heyting algebra with cardinality $C(n)$, where $C(n)$ is the $n$th Catalan number, $$1,1,2,5,14,42,132,\ldots$$ I'm looking for a reference, if this fact is known (to be known).
Below I will explain where the Heyting algebra structure comes from, in case it helps. When $n=0$ or $n=1$, we have $C(n)=1$, and there is a unique Heyting algebra structure on a set with one element, so suppose $n\geq 2$.
For any $m\in\mathbb{N}$, let $[m]:=\{0,1,\ldots,m\}$ and for any $0\leq a\leq b\leq m$, write $[a,b]$ for the subinterval $\{a,a+1,\ldots,b\}\subseteq[m]$. These subintervals form a poset, which we consider as a topological space with the Alexandrov topology: points are subintervals $[a,b]$ and open sets are down-closed subsets. Write $\Omega[m]$ for the poset of open sets in this space, so it has the structure of a Heyting algebra. It remains to show that the cardinality of $\Omega[m]$ is $C(m+2)$.
It is well-known that the Catalan number $C(n)$ counts the Dyck paths of length $2n$. These are paths in a triangle of dots (see below for $n=5$), starting at the southwest point, ending at the northeast point, where each edge in the path moves one unit either northward or eastward.
Position the elments of $[m]$ in the $(m+2)$-triangle, as shown here in the case $m=3$: $$ \begin{array}{ccccccccccc} \bullet&&\bullet&&\bullet&&\bullet&&\bullet&&\bullet\\ &&&&&&&3\\ \bullet&&\bullet&&\bullet&&\bullet&&\bullet\\ &&&&&2\\ \bullet&&\bullet&&\bullet&&\bullet\\ &&&1\\ \bullet&&\bullet&&\bullet\\ &0\\ \bullet&&\bullet\\ \\ \bullet\\\\ \end{array} $$ In this setup, a Dyck path $p$ of length $m+2$ can be identified with a downclosed subset, $S(p)\in\Omega[m]$. For example, the Dyck path $p_0$ shown below $$ \begin{array}{ccccccccccc} \bullet&&\bullet&&\bullet&&\bullet&-&\bullet&-&\bullet\\ &&&&&&|&3\\ \bullet&&\bullet&-&\bullet&-&\bullet&&\bullet\\ &&|&&&2\\ \bullet&&\bullet&&\bullet&&\bullet\\ &&|&1\\ \bullet&&\bullet&&\bullet\\ &0&|\\ \bullet&-&\bullet\\ |\\ \bullet\\\\ \end{array} $$ represents the set $S(p_0)=\mathord{\downarrow}[1,2]\cup\mathord{\downarrow}[3]$.
In fact, all these Heyting algebras $\Omega[m]$ fit together in a single topos, as we now explain. Consider the additive monoid of natural numbers as a category $BN$ with one object. Let $\mathbf{Int}:=\mathrm{Tw}(BN)$ be the twisted arrow category, and consider the presheaf topos $\mathrm{Psh}(\mathbf{Int})$. The subobject classifier for this topos is a functor $$\Omega'\colon\mathbf{Int}^\mathrm{op}\to\mathbf{Set}.$$ so for each object $n\in\mathbb{N}=\mathrm{Ob}(\mathbf{Int})$, we have a set $\Omega'(n)$. Moreover this set carries the structure of a Heyting algebra. Finally, $\Omega'(n)$ has a well-known description in terms of sieves, i.e. subfunctors of the representable functor $\mathbf{Int}(-,n)$. Unwrapping the definition, these are exactly the open sets of $[n]$. In other words, we have a bijection $\Omega'(n)\cong\Omega[n]$.
