Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 160 / 1 / a / ii / Solution

Ina fixed tabloid $\{u\}$ can occur at most once. Indeed, if $\{\pi t\}=\{\sigma t\}$ for $\pi ,\sigma \in C(t)$, then ${\sigma }^{-1}\pi$ belongs to both the row and column stabilizers of $t$. Their intersection is trivial, so $\pi =\sigma$. The coefficient $⟨e(t),\{u\}⟩$ is consequently $0$, $1$, or $-1$.