Generating function for partitions

Statement

Let $p(n)$ be the partition function. Then, as formal power series in $\mathbb{Z}[\![x]\!]$, $$\sum_{n \ge 0} p(n)\, x^n = \prod_{k \ge 1} \frac{1}{1 - x^k}.$$

Remarks

The infinite product makes sense, because each coefficient of the product depends only on finitely many factors. Proof idea: the factor $\frac{1}{1 - x^k} = 1 + x^k + x^{2k} + \dots$ chooses how many parts equal $k$. Restricting the factors gives other results. For example, $\prod_{k \ge 1} (1 + x^k)$ counts partitions into distinct parts. This gives Euler's partition theorem.

Explanations

L1 human

Each factor of the product decides how many times a part $k$ appears in the partition. Multiplying the factors combines these choices, and the coefficient of $x^n$ counts the choices with total $n$.