I know that the expression $\prod_{i \geq 1}\frac{1}{1-x^i} = \sum_np(n)x^n$ counts the number of partitions of $n$. I want to know if we can give a similar interpretation to an expression of the form $\prod_{i\geq 1} \frac{1}{1-tx^i}$. If we can, I would like to know how exactly we can interpret the $t$ part of the expression!
Thanks for your help!
We know that $$\frac{1}{1-tx^i} = 1 + tx^i + t^2x^{2i} + t^3 x^{3i} +\cdots,$$ so the coefficient of $t^kx^n$ in $$\prod_i \frac{1}{1-tx^i}$$ counts the number of partitions of $n$ into exactly $k$ parts. To see this, note that we get exactly one factor of $t$ for each part.
This is also the number of partitions into parts the largest of which is $k$.