Let $\lambda = (\lambda_1, \ldots, \lambda_k) \vdash n$ be a partition. The dimension of the corresponding irreducible representation $V^\lambda$ (equivalently, the number of standard Young tableaux of shape $\lambda$) is: $\dim V^\lambda = f^\lambda = \frac{n!}{\prod_{(i,j) \in \lambda} h_{ij}}$ where $h_{ij}$ is the hook length of the box at position $(i,j)$: $h_{ij} = \lambda_i - j + \lambda_j' - i + 1$ Here $\lambda_j'$ is the size of column $j$ (the transpose partition component).

Equivalently: $h_{ij}$ = (boxes to the right) + (boxes below) + 1.

The sum of squares of dimensions equals the group order: $\sum_{\lambda \vdash n} (f^\lambda)^2 = n!$

This follows from the regular representation decomposition: \mathbb{C}[S_n] \cong \bigoplus_{\lambda \vdash n} V^\lambda^{\oplus f^\lambda}