= Rim-hook length occurs as an ordinary hook length
{title2=$|\lambda/\mu|=h_\lambda(i,\mu_j+1)$}
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 $i\leq l<j$, 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.
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