Let $\pi$ denote a partition of the integer $n$. For the cell in position $(i,j)$ (row $i$ counting from top to bottom, and column $j$, counting from left to right) in the Young diagram for $\pi$, the hook for that cell, $H_{\pi}(i,j)$, is the collection of cells in the same row and to the right of this cell, together with the cells in the same column and below this cell, together with this cell itself. For example the picture below shows the Young diagram for the partition $5+4+2+1$, together with the hook (in green) for the $(1,3)$ cell.

The hook length for the cell $(i,j)$, denoted $h_{\pi}(i,j)$, is the number of cells in the associated hook $H_{\pi}(i,j)$. In the diagram above, it can be seen that the number of cells in the hook for the $(1,3)$ cell is 4, so the hook number for this cell is 4. The picture below shows the hook numbers in all of the cells in the Young diagram for the partition $5+4+2+1$.

The hook-length formula expresses the number of standard Young tableaux of shape $\pi$ associated with the partition $\pi$ of $n$, denoted $d_{\pi}$ can be computed from the hook diagram for $\pi$ as \[ \frac{n!}{\prod_{i,j}h_{\pi}(i,j)}, \] where the product is over all cells $(i,j)$ in the Young diagram for $\pi$. For the partition $5+4+2+1$ above of 12, the number of Young tableaux is \[ \frac{12!}{8\times 6 \times 4 \times 3\times 1 \times 6 \times 4 \times 2 \times 1 \times 3 \times 1 \times 1}=5775. \] A t-core Partitions is a partition such that none of the hook lengths are divisible by $t$.

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