Enumeration of self-complementary structures

Abstract

For functions f : D → R k where D is a finite set and R k = {0,1,… k} we define complementary and self-complementary functions. De Bruijn's generalization of Polya's theorem gives a formula for the number of non-isomorphic self-complementary functions f ∈ R k D . We consider the special cases of generalized graphs and m-placed relations. Among other results we prove that the number of non-isomorphic self-complementary relations over 2 n elements is equal to the number of non-isomorphic self-complementary graphs with 4 n + 1 points.