A small partition-bijection challenge

A small partition-bijection challenge

Suppose that two families of integer partitions, say $\mathcal A_n$ and $\mathcal B_n$, satisfy

or equivalently,

for every $n$ in some range (or for every $n\ge 0$).

The equality may come from an algebraic identity, a recurrence, or an identity between generating functions. My question is:

Can one construct an explicit and natural bijection

Of course, equality of the cardinalities guarantees the existence of some bijection when the sets are finite. The interesting problem is whether the numerical identity can be explained combinatorially by a map preserving meaningful partition statistics.

More generally, if an identity of the form

holds, under what conditions should one expect a bijective or weight-preserving explanation?

I am particularly interested in examples where the equality is easy to prove analytically but finding the corresponding bijection is nontrivial.