I'm trying to understand the calculation for the cycle index of a Symmetric Group $S_n$.
This is discussed and presented in mathematical format on the wikipedia here:
https://en.wikipedia.org/wiki/Cycle_index#The_symmetric_group_Sn
My first question is merely a sanity-check, to make sure I'm understanding what I'm reading: Wikipedia says:
"There are three steps: first partition the set of $n$ labels into subsets, where there are $j_k$ subsets of size $k$."
If I'm on the right track, this $j_k$ is a Stirling number of the first kind, $j_k = Stirling1(n,k)$. Is this correct?
My second question regards the non-recursive presentation for $Z(S_n)$, wikipedia presents:
$\frac{n!}{\prod_{k=1}^n k^{j_k}j_k!}$
If I'm right (above) that $j_k$ is the Stirling number of the first kind for $n,k$ then I can compute this fraction for $n$ and I will get a single number. But I expect that a cycle index $S_n$ is not a single number but a sum of dummy variables. I've obviously gone off track somewhere. Any advice is much appreciated!