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!