Showing two partitions to be equal

Question

I was posed with the following problem " Let $F(n)$ denote the number of partitions of $n$ with every part appearing at least twice and $G(n)$ denote the number of partitions of $n$ into parts larger than 1 such that no two parts are consecutive integers. Use conjugate partitions to prove that $F(n)=G(n).$"

What i have so far:

I realize that for G(n), the statement that no two parts are consecutive implies that distance between two is at least two, which gives a similarity with F(n). I think using ferrer diagrams will help a bit, but i could not see how to get started. Please help

Answer 1

Here's the Ferrers diagram for $n=6$.

I realize that for $G(n)$, the statement that no two parts are consecutive implies that distance between two is at least two,

This might be causing a problem. No two parts are consecutive implies the difference between two parts is not $1$ instead of at least two as you have stated.