= Solution
Trace the southeast boundary of the <Hook of a Young diagram> based at $(i,j)$. At each horizontal boundary step record the hook length $h_{i,y}$ of the cell in row $i$ above that step. At each vertical step ending beside row $x$, record $h_{i,j}-h_{x,j}$. Starting at the northeast end and moving to the southwest end, these records increase by one from $1$ to $h_{i,j}$; horizontal and vertical steps are disjoint and account for every step. Therefore the <Hook-interval decomposition at a Young-diagram cell> is
$$
\boxed{
\{1,\ldots,h_{i,j}\}
=\{h_{i,y}:j\leq y\leq\lambda_i\}
\sqcup
\{h_{i,j}-h_{x,j}:i<x\leq\lambda'_j\}.}
$$
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