Hook of a Young diagram

The hook $H_{i,j}(\lambda )$ consists of $(i,j)$, the cells to its right in row $i$, and the cells below it in column $j$. Its hook length is

For $(i,j)\in Y(\lambda )$, the integers from one through $h_{i,j}$ split as the disjoint unionTracing the southeast boundary of the hook records the first set at horizontal steps and the second at vertical steps.

If a partition has a hook of length $ef$, it has a hook of length $e$. In a beta set, the first hook is a bead-gap pair at distance $ef$; along the arithmetic progression with step $e$, some consecutive pair changes from a bead to a gap.

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.