Some other ways to prove a partitions problem

Question

Show that the number of partitions of the integer $n$ into three parts equals the number of partitions of $2n$ into three parts of size $< n$.

I can only prove it by building a bijection between the two sets. Could anyone prove it by generating functions or even by Ferrers diagram?

Answer 1

I am not sure of what you mean "by Ferrers diagram"

But take three rows of $n$ dots and put a partition of the first type in; the remainder will represent a (rotated) partition of the second type. And similarly in reverse. For example with $n=6$, one partition of the first type is $12=5+5+2$ and of the second type is $6=4+1+1$, as shown here.

Answer 2

This may be what you’ve already done, but you can use Ferrers diagrams to produce a bijection that pairs a partition $P$ of $n$ with the partition of $2n$ whose Ferrers diagram, rotated 180°, fits together with the Ferrers diagram of $P$ to form a $3\times n$ rectangle. For example, for $n=6$: $$\left[\begin{array}{l|r}oo&oooo\\oo&oooo\\oo&oooo\end{array}\right];\left[\begin{array}{l|r}ooo&ooo\\oo&oooo\\o&ooooo\end{array}\right];\left[\begin{array}{l|r}oooo&oo\\o&ooooo\\o&ooooo\end{array}\right]$$