I am aware that the number of necklaces with $m$ red beads and $n$ white beads, where $\gcd(m,n)=1$, is equal to $\frac{1}{m+n}\binom{m+n}{n}$. For example, if $m = 3$ and $n = 4$, there are 5 possible necklaces.
I am curious as to whether there is a “natural” cycle structure on the set of red/white necklaces. In the previous example, such a cycle might begin
RRRWWWW $\to$ RRWRWWW $\to$ RRWWRWW $\to \dots,$
where the intent is to switch the positions of some of the beads until we’ve seen every possible necklaces exactly once. Ideally this cycle procedure would be describable for different numbers of beads. Thank you in advance for your help.