Hook-interval decomposition at a Young-diagram cell
= Hook-interval decomposition at a Young-diagram cell
{c}
For $(i,j)\in Y(\lambda)$, the integers from one through $h_{i,j}$ split as the disjoint union
$$
\{h_{i,y}:j\leq y\leq\lambda_i\}
\sqcup
\{h_{i,j}-h_{x,j}:i<x\leq\lambda'_j\}.
$$
Tracing the southeast boundary of the hook records the first set at horizontal steps and the second at vertical steps.