Staircase-character vanishing criterion

For a partition $\lambda$, the irreducible character ${\chi }^{\lambda }$ vanishes on every cycle type containing an even part exactly when $\lambda$ is a staircase $(m,m-1,\dots ,1)$. A staircase has only odd hook lengths. Conversely, vanishing first forces $\lambda$ to be self-conjugate; applying the Murnaghan–Nakayama rule at the largest even hooks then forces consecutive row lengths.