Let $N$ be $\{1,\ldots,n\}$, $R=\{1,\ldots,r\}$. Consider mappings from $N$ to $R$. It is clear that there are $r^n$ such mappings.
Let $G_n$ be the group generated by $n$ length cycle permutation $\alpha=(1,\ldots,n)$. $G_n$ contains all powers of $\alpha$.
One can see that the group $G_n$ acts on the set of mappings in the following way. Let $\alpha^k\in G_n$ and $f:R\mapsto N$.
$\alpha^k(f)$ would be the following mapping $(\alpha^k(f))(i)=f(\alpha^k(i))$.
We can use Bernside's lemma to find number of orbits. Now I need to find algorithm to find orbits. By finding orbits I mean finding set or representatives from each orbit.
Here is an example:
$n=4, r=2$.
A mapping from $\{1,2,3,4\}$ to $\{1,2\}$ can be represented by a vector $(f(1),f(2),f(3),f(4))$.
Group $G_4$ would be $\{(1234),(13)(24),(1342),e\}$
One can see that there are following $6$ orbits:
$\{(1,1,1,1)\}$
$\{(2,2,2,2)\}$
$\{(1,1,1,2),(1,1,2,1),(1,2,1,1),(2,1,1,1)\}$
$\{(2,2,2,1),(2,2,1,2),(2,1,2,2),(1,2,2,2)\}$
$\{(2,2,1,1),(2,1,1,2),(1,1,2,2),(1,2,2,1)\}$
$\{(2,1,2,1),(1,2,1,2)\}$
A set of representatives would be the first column.
As a algorithm one can choose the following. Mark all mappings unchecked. At each step get one unchecked mapping, put it in representation set, act group $G_n$ on it, get all resulting mappings and make them checked. Continue until all elements are checked.
I have posted general question for this problem here https://mathoverflow.net/questions/325839/find-representation-set-of-orbits-when-group-acts-on-a-set
I think there should be better algorithm at list for this specific case. Can anyone help?