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Put $L=h_{i,j}=pr$. The Hook-interval decomposition at a Young-diagram cell is the disjoint unionHere is a proof of the identity using the hook criterion in a beta set. Keep only the cells weakly southeast of $(i,j)$, translating that cell to $(1,1)$; this gives the partition of an integerLet $\ell ={\lambda }_j^{\prime }-i+1$ and use its beta set of a partition $b_s={\nu }_s+\ell -s$. These numbers are precisely the hook lengths down the original column from $(i,j)$, with $b_1=L$. The hooks along the first row of $\nu$ are the differences $L-c$ for the gaps $0\le c<L$. The remaining positions in that interval are the beads $b_2,\dots ,b_{\ell }$, giving exactly the displayed disjoint decomposition. The first-row and first-column hook lengths of $\nu$ agree with those in the original Young-diagram hook.

Because $p\mid L$, a leg cell satisfies $p\mid h_{x,j}$ exactly when $p\mid L-h_{x,j}$. Thus divisibility by $p$ identifies the relevant cells of the Young-diagram hook with the multiples of $p$ in $\{1,\dots ,pr\}$. There are exactly $r$ of those, provingThe count includes the corner cell $(i,j)$ itself.

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