Enumeration of self-complementary structures

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 $\text{f ∈ R}{}_{\mathrm{k}}{}^{\mathrm{D}}$. 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 2n elements is equal to the number of non-isomorphic self-complementary graphs with 4n + 1 points.