Given a finite set S of size n, let E denote the set of its subsets with even size and let O denote the set of its subsets of odd size. Then it's easy to see that #(E) = #(O). (Expanding the LHS of (1-1)^n = 0 does the trick nicely.) When n is itself odd, there is a simple natural bijection between E and O: just map any subset of S to its complement. But when n is even, there doesn't seem to be anything nearly so natural. (If we set S = {1,...,n} and each subset X = {a,b,...,c} where a < b < ... < c, then one can certainly list the elements of E by size and, within each size, by lexicographic ordering, and likewise for O. This defines a bijection E -> O. But that's less natural than just taking each subset to its complement.) QUESTION: Exactly how is "natural" defined so that complementation is a natural bijection E -> O for n odd, but there is no known natural bijection E -> O for the case of n even? Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele