Past exam of the Lie algebras and representation theory of symmetric groups course of the University of Oxford / 2007 / B / 5 / d / Solution

For shape $(n-2,2)$ with $n\ge 6$, there are exactly two removable rim hooks of size two: the bottom row, leaving $(n-2)$, and the final two cells of the first row, leaving $(n-4,2)$. Both have height zero. A vertical pair cannot be removed while leaving a Young diagram because the first row is longer than the second by at least two.

By the Murnaghan–Nakayama rule and part (b),