An Arithmetic View on Generating Necklaces

A k-ary n-tuple necklace is an equivalence class of k-ary strings under rotation. The classical idea is to treat them as characters. As the numbers being treated as characters can always being treated as numbers, like we treat char and int as the same type of variables in computer science, we present a new arithmetic algorithm using the decomposition of the ring Z/(kⁿ - 1) on generating necklaces. All the k-ary necklaces of length n can be generated in a straight and simple way with no prenecklaces or other redundancies by our method.