Hook-length formula

Statement
Suppose $$\lambda$$ is an unordered integer partition of a natural number $$n$$. We can describe $$\lambda$$ by means of a Young diagram (also called a Ferrers diagram).

For every box in this Young diagram, define the hook-length at that box as follows:

Hook-length = 1 + (number of boxes directly below it vertically) + (number of boxes directly to the right of it horizontally)

Then, we have the following:

Degree of irreducible representation corresponding to partition $$\lambda$$ = Number of (standard) Young tableaux with shape $$\lambda$$ = Number of paths in the Young lattice from the trivial partition of $$1$$ to $$\lambda$$ = $$\frac{n!}{\prod \operatorname{hook-lengths}}$$

The product in the denominator is the product, over all boxes in the Young diagram, of the hook-length corresponding to that box.