Infinite product expression of partition function

Question

I'm working on a problem (specifically, I'm using an exam paper without course notes to prepare for a course starting in September),

Define the partition function $P(q)$ and give its infinite product expression.

Wikipedia has

The generating function for $p(n)$ is given by $$\sum_{n=0}^\infty p(n)x^n=\prod_{k=1}^\infty\left(\frac{1}{1-x^k}\right)$$

But how does this allow us to evalute $p(n)$?

Answer 1

From comments as requested:

$\frac{1}{1-x^k} = 1+x^k+x^{2k}+x^{3k}+\cdots$.

As an example, to find the number of partitions of $0$, $1$, $2$, $3$, and $4$, expand $(1+x+x^2+x^3+x^4+\cdots)(1+x^2+x^4+\cdots)(1+x^3+\cdots)(1+x^4+\cdots)(1+\cdots)$ to get $1+x+2x^2+3x^3+5x^4+\cdots$ so the answers are $1$, $1$, $2$, $3$, and $5$ respectively.

• The formal way to use generating functions: take the $n$th derivative of the sum or product at $x=0$ and divide that by $n!$; clearly with the sum that gives $p(n)$, and doing it to the product gives the numerical value.

• Easy method: look at the coefficient of $x^n$ in the expansion.