Proof of an integer partitions inequality

Question

I came across an interesting problem the other day.

Let $P_n$ be the number of partitions of a positive integer $n$. For instance $P_4$ = $5$, as there are five ways of partitioning $4$:

• $4$
• $3+1$
• $2+2$
• $2+1+1$
• $1+1+1+1$

Prove that $P_n$ < $\sqrt{P_{n(n+2)}}$.

The way I tried to prove this is by bounding $P_n$ from above by some function $F(n)$ and bounding $P_{n(n+2)}$ from below by some function $G(n)$ such that $(F(n))^2$ < $G(n)$. Unfortunately, I wasn't able to find bounds that would satisfy the inequality. How to go about proving this?

Answer 1

You want to prove that $P_n^2<P_{n^2+2n}$. Consider Ferrers diagrams. Start with an $n$-by-$n$ square, and put Ferrers diagrams of partitions of $n$ to the right and below.