USAMO 1986 Problem 5

A partition of $n$ is an increasing sequence of integers with sum $n$. For example, the partitions of 5 are: $1,1,1$, 1,$1 ; 1,1,1,2 ; 1,1,3 ; 1,4 ; 5 ; 1,2,2 ;$ and 2,3 . If $\mathrm{p}$ is a partition, $\mathrm{f}(\mathrm{p})=$ the number of $1 \mathrm{~s}$ in $\mathrm{p}$, and $\mathrm{g}(\mathrm{p})=$ the number of distinct integers in the partition. Show that $\sum f(p)=\sum g(p)$, where the sum is taken over all partitions of $n$.