Combinatorics: is there a reasonably straightforward explicit formula for what I'm denoting $a^b_\mathcal{P}$?

Question

Given natural numbers $a$ and $b$, define $a^b_\mathcal{P}$ as follows. Let $A$ denote a set with $a$ elements and $B$ denote a set with $B$ elements. Then $a^b_\mathcal{P}$ is the number of monotone (i.e. order-preserving) maps $\mathcal{P}(A) \rightarrow \mathcal{P}(B)$. For example, $a^1_\mathcal{P}$ equals the number of uppersets in $\mathcal{P}(A),$ for $|A|=a$.

Question. Is there a reasonably straightforward explicit formula for what I'm denoting $a^b_\mathcal{P}$?

I'd also be interested in terminology, standard notation, further references etc.

Answer 1

This question has an easy answer, which is no. Already the simplest example is a notoriously difficult to compute sequence, the Dedekind numbers. One would guess there's little hope for the general case.