Determining the basis for strings with cyclic and permutation symmetry I'd like to determine a basis set of strings for strings of length m composed of n letters that are compatible with cyclic symmetry of the string and permutation symmetry of the letters.
It's relatively straightforward to do this by hand in individual cases. For example $m=5, n=3$ with the letters labelled $1,2,3$, then the minimal number of strings I believe are
$$ {00000}, {00001},{00011},{00101},{00012},{00112},{00102}, {00121}, {01012}$$
where I can generated all the other strings by cyclically permuting the indices of the strings above, or by permuting the letters {1,2,3} amongst themselves.
Is there are a way of determining the number of basis strings for each $m,n$? Or at least the number of basis strings constructed in this way.

The problem described here seems to be identical to the problem of finding the number of equivalence classes of base-m necklaces of length n, where necklaces are considered equivalent under both rotations as well as permutations of the symbols.
The solution is given in N. J. Fine, Classes of periodic sequences, Illinois J. Math., 2 (1958), 285-302. 
