I'm working on the following exercise:
For $n\ge 7$, $S_n$ has no irreducible representations of dimension $m$ with $2\le m\le n-2$.
There is a solution here but I'd like to follow the suggestion of Fulton and Harris (exercise 4.14) and prove this using the hook-length formula. Presumably the proof will proceed inductively, and I've shown the base case $n=7$, but it's the rest I'm struggling with.