= Hook length
{title2=$h_{i,j}(\lambda)$}
The hook length of cell $(i,j)$ in the <Young diagram> of a <partition of an integer> $\lambda$ is the number of cells in its <Young-diagram hook>:
$$
h_{i,j}(\lambda)=\lambda_i-j+\lambda'_j-i+1.
$$
Here $\lambda'$ is the <conjugate partition>. There is one hook length for every cell, so the <multiset> of hook lengths has <cardinality> $|\lambda|$.
For $\lambda=(3,2)$ and $\lambda'=(2,2,1)$, compute $h_{1,1}=(3-1)+(2-1)+1=4$ and $h_{1,2}=(3-2)+(2-1)+1=3$. Continuing cell by cell gives the <hook graph of a partition>
$$
\begin{array}{ccc}4&3&1\\2&1&\end{array}
$$
or the row array $[[4,3,1],[2,1]]$. Its <hook product> is $4\cdot3\cdot1\cdot2\cdot1=24$. A cell has <hook length> one exactly when it is a <Removable node of a Young diagram>: it has no cell to its right or below. Here these are $(1,3)$ and $(2,2)$. Transposing the <Young diagram> gives the array $[[4,2],[3,1],[1]]$ with the same <hook product>; this explains geometrically why conjugate shapes have equal counts in the <hook-length formula>.
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