A few days ago I asked this question on a generating function of multiset cycles.
There is was shown that $\prod_C(1-w(C))^{-1}=\sum_{\pi}w(\pi)$ where $w(\pi)$ is the weight a a multiset permutation $\pi$ as defined in the previous question. There's something more I'm curious about.
Supposedly it's the case that if $p_k=x^k_1+x^k_2+\cdots$, then $ \prod_C(1-w(C))^{-1}=\prod_{k\geq 1}(1-p_k)^{-1}. $
The explanation is that one notes that $p^n_k=\sum_{d\mid n}\sum_A d(w(A))^{nk/d}, \tag{*}$ where $A$ ranges over all aperiodic cycles of length $d$, that is, cycles of length $d$ unequal to a proper cyclic shift themselves. Substituting this equality into the expansion of $\log\prod(1-p_k)^{-1}$ gives the desired result upon simplification.
I don't fully follow this terse explanation. Can someone further clarify how the identity $(*)$ is deduced, and why the substitution yields the result? Thank you graciously.