# Prove that number of partitions $n^2$ on $n$ distinct parts $=$ partitions of $\binom{n}{2}$ on at most $n$ parts

Prove that number of partitions $$n^2$$ on $$n$$ distinct parts $$=$$ partitions of $$\binom{n}{2}$$ on at most $$n$$ parts

I tried to find bijection. We can noticed that $$\binom{n}{2} = \frac{n^2-n}{2}$$ which is kind of hint there...
Let find bijection in use of example. Let $$n=8$$ and $$n^2 = 64$$ let $$\pi$$ be $$1+2+3+4+5+6+7+8+9+19$$ Now rearrange them: Odd first, even later: $$2+4+6+8+1+3+5+7+9+19$$ now we do $$n^2 \rightarrow n^2-n$$: $$1+3+5+7+0+2+4+6+8+18$$ Now do $$/2$$: $$0+1+2+3+0+1+2+3+4+9$$

OK but there are two problems