A permutation group A of degree n acting on a set X has a certain number of orbits, each a subset of X. More generally, A also induces an equivalence relation on X(k), the set of all k-subsets of X, and the resulting equivalence classes are called k-orbits of A, or generalized orbits. A self-complementary k-orbit is one in which for every k-subset S in it, X-S is also in it. Our main results are two formulas for the number s(A) of self-complementary generalized orbits of an arbitrary permutation group A in terms of its cycle index. We show that self-complementary graphs, digraphs, and relations provide special classes of self-complementary generalized orbits. (Author)
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