This is probably very elementary, but is there a way to get the number of equivalent (not necessarily ordered) partitions of an ordered $k$-fold partition $p=(p_1,..,p_k)$ of $n \in \mathbb{N}$ without knowing how often the $p_i$'s appear?
Example: $n \in \mathbb{N}$, $k=2$. In this case, the formula $$|\bar{p}|=\lim_{q \to 0}\, (1+(1-q^{p_2-p_1}))$$ does the job. Here, $\bar{p}$ denotes the equivalence class of $p$ and $|\,.\,|$ is the cardinality. The two summands represent the elements $(e,\sigma_1)$ in $S_2$, which gives an idea how to generalize this..