Let $\sigma$ be an element of $SL\_{24}(\mathbb{Z})$ with its Jordan normal form is diagonal and the eigen values are $\epsilon\_j$ for $1 \le j \le 24$ are n th root of unity where $n|N$ and $N$ is the finite order of $\sigma$. Equivalently we are describing $\sigma$ through its cycle shape $(a\_1)^{b\_1}\cdots(a\_s)^{b\_s}$. 


We associate $\sigma$ to the following modular form: 


$$\eta\_{\sigma}(q) := \eta(\epsilon\_1q)\cdots\eta (\epsilon\_{24} q)=\eta(q^{a\_1})^{b\_1}\cdots \eta(q^{a\_s})^{b\_s}$$


Here $\eta$ stands for the Dedekind eta-function. Using the above defined $\eta\_{\sigma}$ we define: 
$$\sum\_{j>0}p\_{\sigma}(1+j)q^{1+j} = \frac{q}{\eta\_{\sigma}(q)}$$
This is a generalized partition function.


We assume the cycle type of $\sigma$ is $1^{1}23^{1}$ and hence $N = 23$. In this case I have the following questions :


1. What is this generalized partition function $p\_{\sigma}$ and how to find $p\_{\sigma}(n)$ for some natural number $n$?
2. There are many generalisations of partitions functions and hence what is the reference for this particular type of generalized partition function?


Thanks for your time.


Have a good day.



Generalized partitions and eta functions
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. For example, in your case where $b\_1=b\_2=a\_1=1, a\_2=23,$ 
$$f(q):=\frac1{\prod\_{k>0}(1-q^k)(1-q^{23k})}=1+q+2q^2+3q^3+5q^4+\dots$$
which is the generating function of partitions of $n$ into positive integers where the integers come in two kinds and the second kind has weight $23$ times the weight of the first kind.


