Past exam of the Lie algebras and representation theory of symmetric groups course of the University of Oxford / 2009 / B / 4 / 1 / Solution

The Young diagram is $[\lambda ]=\{(i,j):1\le i\le k,1\le j\le {\lambda }_i\}$. The hook length of a box iscounting the box itself, the boxes to its right in the same row, and the boxes below it in the same column. The hook-length formula isThe first implication in the equivalence is immediate: if every hook length is odd, no hook can have length two.