Let $p_k(n)$ denote the number of partitions of the positive integer $n$ into at most $k$ parts. I am trying to answer a question which asks me to show that the generating function for the sequence $\{ p_k(n) \}$ is given by $$ P_k(n) := \frac{1}{\prod_{r=1}^k (1-x^r)} $$
Can anyone help me to understand how this might be done?
Recall that $$\frac{1}{1-x^a}=\sum _{i = 0}^{\infty}x^{ai},$$ hence the product looks like $$(1+x+x^2+\cdots +\color{red}{x^{x_1}}+\cdots )(1+x^2+x^4+\cdots +\color{red}{x^{2x_2}}+\cdots )\cdots (1+x^k+x^{2k}+\cdots +\color{red}{x^{kx_k}}+\cdots)$$ so when you multiply together you are picking the $x_1,x_2,\cdots ,x_k$ in each part and so the coefficient that you ended up with $x^n$ is $$\sum _{1*x_1+2x_2+\cdots kx_k=n}1=|\{(x_1,\cdots ,x_k):\sum _{i = 1}^ki*x_i=\underbrace{1+1+\cdots +1}_{x_1}+\underbrace{2+\cdots +2}_{x_2}+\cdots +\underbrace{k+\cdots +k}_{x_k}=n\}|=p_k(n)$$
