Consider the partition $\displaystyle \lambda=(m,n-m)$ of $\displaystyle n$ (thus $\displaystyle 2m \ge n$.)


The number of Young
tableaux of shape $\displaystyle \lambda$ is given by
$\displaystyle f_{(m,n-m)} = \binom nm - \binom{n}{m+1}$


a) Prove this using the hook-length formula.


b) Prove this using induction on $\displaystyle n$

Please, help!