Solution
= Solution
In
$$
e(t)=\sum_{\pi\in C(t)}\operatorname{sgn}(\pi)\{\pi t\},
$$
a 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 $\langle e(t),\{u\}\rangle$ is consequently $0$, $1$, or $-1$.