The partition bijection question originally asked by Wouter left open the possibility that might be no "natural" bijection between these two kinds of partitions.

I guess the quintessential example is, given an arbitrary vector space V (e.g., say V is a 1-dimensional v.s. over R), there is no "natural" bijection between V and V* (= linear maps from V to R).

I can think of two examples from combinatorics in which one can demonstrate that there is no "natural" bijection between two sets of the same size. #1: The set of linear orderings of an n-element set S has n! elements, as does the set of invertible self-maps from S to itself. And one can put the two sets in bijection if one specifies a special linear ordering of the elements of S. But there is no "natural" bijection, if by a natural bijection one means a G-equivariant bijection, where G is the group of permutations of S. After all, there is a very special self-map from S to itself, namely the identity map, but there is no privileged linear ordering of a set S that comes with no extra structure. (Putting things in the language of G-equivariance: the G-set of self-maps has a fixed point under the action of G, while the G-set of linear orderings does not.) #2: If S is an n-element set with n>0 even, S has 2^{n-1} subsets of even cardinality and it also has 2^{n-1} subsets of odd cardinality. And one can put the two collections of subsets in bijection (indeed, if one views the power set of S as an n-dimensional vector space over the field with 2 elements, then a linear map will do). But there is again no G-equivariant bijection. After all, there is a very special even-cardinality-subset of S, namely S itself (or, if you prefer, the empty set), but there is no privileged odd-cardinality-subset of S. If you like thinking about things this way, you might enjoy the article "Producing New Bijections from Old" by David Feldman and myself (published in Advances in Mathematics, volume 113 (1995), pages 1-44), which unfortunately was written just before I started putting all my work on the web. Jim