# Maps between combinatorial necklaces

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 the problem I am trying to solve, I need to find a series of maps that can be used to transform one possible necklace into any other necklace that has $m$ red and $n$ white beads. For example, if $m = 3$ and $n = 4$, there are 5 possible necklaces. Do maps exist that can send RRRWWWW to RRWRWWW, RRWRWWW to RRWWRWW, and so forth? The map that takes necklace 1 into necklace 2 does not necessarily have to be the same as the map that takes necklace 1 into necklace 3. Thank you in advance for your help.