Rim hook

A rim hook, or border strip, is a connected skew Young diagram containing no $2\times 2$ square. Its leg length is one less than its number of occupied rows.

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.

For a type $\alpha$, each successive difference is a rim hook of size ${\alpha }_i$; zero parts give empty differences of height zero. Label its cells by $i$. Such fillings are weakly increasing along rows and columns, and the height is the sum of the occupied strips' numbers of rows minus one. Empty parts are omitted from the associated product of positive power sums; the finite-variable degree-zero power sum is not inserted.