The number of self-complementary achiral necklaces

Necklaces with beads of two colors which are left unchanged both by a reflection as well as by the interchange of the two colors are characterized in terms of their axes of symmetry. This characterization is then used to enumerate them. For n = 2^rm with r ≥ 1 and m odd, the number of self-complementary achiral necklaces is for large even n, this sum is asymptotic to 2^{n/4}-1.