Self-Complementary Generalized Orbits of a Permutation Group

Abstract

AbstractA permutation groupAof degree n acting on a setXhas a certain number of orbits, each a subset ofX. More generally, A also induces an equivalence relation onX(k)the set of allksubsets ofX, and the resulting equivalence classes are calledkorbits ofA, orgeneralized orbits. Aself-complementary k-orbitis one in which for everyk-subsetSin it,X—Sis also in it. Our main results are two formulas for the numbers(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.