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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.

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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. You may find some similar kinds of partitions in the OEIS. For example, sequence 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.

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It is not an answer, but just some comments where such $\eta$ products appear:

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  • $\begingroup$ Great answer. I am excited to know there is this much literature behind this question. Thanks. $\endgroup$ – GA316 Oct 1 '17 at 13:09

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