Solution
= Solution
Choose $g$ whose <disjoint permutation cycles> have lengths equal to the <principal hook lengths> of $\lambda$. Those lengths are distinct and sum to $n$, so this is a permutation in $S_n$. In the iterated <Murnaghan–Nakayama rule>, there is a unique complete sequence that removes the corresponding principal rim hooks. Its contribution is one sign, and hence the <Principal-hook character value of a symmetric group> gives $\chi^\lambda(g)\in\{1,-1\}$.