I have encountered a problem that I suspect has been thoroughly studied but I have not been able to find references. Can anyone point me to a published reference dealing with this or a closely related problem?
How many orbits does the action of the symmetric group of l elements ($\Sigma_{l}$) on $(\mathbb{Z}/n)^{l}$ have? n does not need to be prime.
So in other words:
How many elements does $(\mathbb{Z}/n)^{l}/\Sigma_{l}$ have?
Each orbit contains exactly one element $(a_1,\dots,a_l)$ such that $0\leq a_1\leq a_2\leq \cdots \leq a_l\leq n-1$ (regarding the elements of $\mathbb{Z}/n$ as ordinary integers $0,1,\dots,n-1$). Hence the number of orbits is the number of such sequences, which by elementary, standard reasoning is ${n+l-1\choose l}$.