It is known that a necklace made of two color beads (say $2a$ black ones and $2b$ white ones) can be split into equal portions by two cuts. My question is that given some configuration of such a necklace, how many different fair splits can one have?
Is it related to the necklace counting problem, i.e. maybe the number of ways depends on the size of each necklace equivalence class?
Clearly, if the configuration has a cyclic symmetry $C_n$(n as large as possible), then the answer is $1$ if $n$ is odd, and $2(a+b)/n$ if $n$ is even. However, when there is no cyclic symmetry, it seems one can have many possibilities, and the answer is not so obvious. Are there some results in combinatorics that are relevant?