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). Although this must be true, I've never seen it expressed as a theorem. Can it be? E.g., "Every vector space has a basis" is easily equivalent to the Axiom of Choice via Zorn's Lemma. But is it accurate to say that without AC, there is no bijection from V to V* ? (I suspect one can prove that w/o AC, there is no "natural transformation from the category of all vector spaces to the category of all dual vector spaces" -- or something like that. But what about merely one bijection from V to V* where V is a 1-dim v.s. over R ?) AND, are there countable examples of cases like the two kinds of partitions that don't -- in some sense -- have a natural bijection (more accurately a family of bijections, one for each N) ??? (Imitating the v.s. case, I suppose it's fair to say that w/o AC there's no bijection between G and Hom(G, H), where G is a gp. isomorphic to Z+Z and H is a gp. isomorphic to Z. True? And what about Hom(H,H) ?) --Dan