What you have is positive integers such that $\;a_1b_1+\dots+a_sb_s=24.$
Your $q/\eta_\sigma(q)$ is an example of an eta-quotient and is a modular function of negative weight. As just one example, if $\;a_1=1,b_1=24\;$ then $\eta_\sigma(q)=\Delta(q)$ is the generating function of the [Ramanujan tau function](https://en.wikipedia.org/wiki/Ramanujan_tau_function). You may find some similar kinds of partitions in the [OEIS](http://oeis.org). For example, sequence [A005758](http://oeis.org/A005758) is partitions into parts of 12 kinds. For your first question, just expand $\;\eta(q^a)=\prod_{n>0}1-q^{an}\;$ and multiply the $q$-series together.