Systematic generation of all nonequivalent closest-packed stacking sequences of length N using group theory

An algorithm has been developed that generates all of the nonequivalent closest-packed stacking sequences of length N. There are 2^N + 2(−1)^N different labels for closest-packed stacking sequences of length N using the standard A, B, C notation. These labels are generated using an ordered binary tree. As different labels can describe identical structures, we have derived a generalized symmetry group, Q ≃ D_N × S₃, to sort these into crystallographic equivalence classes. This problem is shown to be a constrained version of the classic three-colored necklace problem.