Rim-hook length occurs as an ordinary hook length ($|\lambda /\mu |=h_{\lambda }(i,{\mu }_j+1)$)

Let a removable rim hook occupy rows $i$ through $j$. Connectedness and absence of a two-by-two square imply that its pieces in consecutive rows overlap in exactly one column. Therefore ${\mu }_l={\lambda }_{l+1}-1$ for $i\le l<j$, and summing the row lengths gives $|\lambda /\mu |={\lambda }_i-{\mu }_j+j-i$. The column $c={\mu }_j+1$ has height $j$, since lower rows are unchanged and have length at most ${\mu }_j$. The hook length at $(i,c)$ is exactly that sum. Thus a diagram with only odd hook lengths has no removable even rim hook. Applying the Murnaghan–Nakayama rule shows that its ordinary character vanishes on every cycle type containing an even part.