# Hook-length formula

From Groupprops

## Statement

Suppose is an unordered integer partition of a natural number . We can describe 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 = Number of (standard) Young tableaux with shape = Number of paths in the Young lattice from the trivial partition of to =

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

## Particular cases

Young diagram | Hook lengths | Hook-length formula | Result of formula | Cross-check | ||
---|---|---|---|---|---|---|

1 | 1 | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] |
1 (for unique box) | 1 | unique Young tableau: fill box with | |

2 | 2 | 2 (for left box), 1 (for right box) | 1 | unique Young tableau: 1 in left box, 2 in right box | ||

2 | 1 + 1 | 2 (for top box), 1 (for bottom box) | 1 | unique Young tableau: 1 in top box, 2 in bottom box | ||

3 | 3 | 3 (for left box), 2 (for middle box), 1 (for right box) | 1 | unique Young tableau: 1 in left, 2 in middle, 3 in right box | ||

3 | 1 + 1 + 1 | 3 (for top box), 2 (for middle box), 1 (for right box) | 1 | unique Young tableau: 1 in top, 2 in middle, 3 in bottom box | ||

3 | 2 + 1 | 3 (for top left box), 1 (for right box), 1 (for bottom box) | 2 | Two possibilities for the Young tableau: 1 in top left, 2 in right, 3 in bottom 1 in top left, 3 in right, 2 in bottom | ||

4 | 4 | 4,3,2,1 as we move from left-most to right-most box | 1 | unique Young tableau: 1,2,3,4 filled left to right |