The following combinatorial problem has bothered me quite a bit. I guess people smarter than me have given the problem some taught as the problem has obvious applications (e.g. to the Ising model), but I have not found any solution on the web (this might be because I don't know the proper terminology).
Anyways, here is the problem:
Consider a string of $N$ binary variables, $\uparrow$ and $\downarrow$. The string will have $2^N$ different configurations. Now impose a symmetry to the system; two configurations are equal if you can get from one to the other by cyclic permutation or by reversal of the string (or a combination of these two symmetries). How many unique configurations will the string have?
For 1 $\uparrow$ and $N-1$ $\downarrow$ there will only be 1 unique configuration. For 2 $\uparrow$ and $N-2$ $\downarrow$ there will be $N/2$ configurations if $N$ is even and $(N-1)/2$ configurations if $N$ is odd. But if you take 3 $\uparrow$ and $N-3$ $\downarrow$, it is no longer clear (at least not to me), how one efficiently should count the number of possible configurations.
I would really appreciated some help, or references on relevant literature.