Exercise 21, § 5

Let $p(n)$ be the number of conjugacy classes of the symmetric group $\mathfrak{S}_n$. Show that $p(n)$ is equal to the number of families $(x_1, \ldots, x_n)$ of integers $\geq 0$ such that $\sum_{i=1}^n i.x_i = n$.

Deduce the identity $\sum_{n=0}^\infty p(n)T^n = \prod_{m=1}^\infty \frac{1}{1 - T^m}$