Genereting function

Question

Let $p(n)$ denote the number of unrestricted partitions of $n$.

How do I explain that the generating function $f(x)$ of $p(n)$ is

$$f(x)=\prod_{n=1}^\infty \displaystyle\frac{1}{1-x^n} $$

Thanks a lot!

Answer 1

A partition is considered to be unordered, so you only need to consider how many elements of the partition have size $n$ for each $n$. When you compute the expansion of $f(x)$ by taking products, you get term $x^M$ precisely when $M = \sum_n k_n n$ for non-negative integers $k_n$ that count how many times $n$ appears in the partition. Thus the coefficient of $x^M$ counts how many partitions give you $M$, as desired.