Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 160 / 2 / b / ii / Solution

For the staircase $\delta =(m,m-1,\dots ,1)$,whenever $(i,j)$ is a cell, so every hook length is odd. A removable rim hook of length $k$ corresponds to a hook of length $k$ in the original diagram. If a cycle type $\mu$ contains an even part $k$, apply the Murnaghan–Nakayama rule to that part first. There are no terms in the sum, and therefore ${\chi }^{\delta }(\mu )=0$.