Young diagram

The Young diagram of $\lambda$ is the set of cells $(i,j)$ with $1\le j\le {\lambda }_i$.

A removable node is a cell whose deletion leaves a Young diagram. The set ${\lambda }^{-}$ consists of the distinct partitions obtained by deleting one removable node from $\lambda$.

The conjugate partition ${\lambda }^{\prime }$ is obtained by reflecting the Young diagram across its main diagonal; thus ${\lambda }_j^{\prime }$ is the number of parts of $\lambda$ at least $j$.

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.

The content of the cell $(i,j)$ is $c_{i,j}=j-i$. Transposing the diagram negates every content and preserves every hook length.

The principal hook lengths are $h_{1,1},h_{2,2},\dots$ along the main diagonal. They are strictly decreasing positive integers, and their sum is the size of the partition.