Symmetric-group character co-degree vanishing criterion

For $\chi \in \mathrm{Irr}(S_n)$, if the order of $g\in S_n$ does not divide $|S_n|/\chi (1)$, then $\chi (g)=0$. The Hook-length formula turns the co-degree into a product of hook lengths, while the Murnaghan–Nakayama rule detects a cycle whose required prime-power hook cannot be removed.