Hook Length Theorem

Introduction

The hook length theorem is a theorem to be used on Standard Young Tableau. A standard Young Tableau is essentially a pack of blocks together, such as this one:

A tableau has a decreasing(not strictly decreasing) number of blocks in every row.

A Standard Young Tableau(SYT) has increasing numbers in both rows and columns. As shown in the figure, 1-3-10 is increasing, as well as 2-5, 4-6, 7-9, 1-2-4-7-8, and 3-5-6-9.

Theorem

Let the number of blocks in the tableau be . A hook of a block is the number of blocks to the right and below the block, including the block.

The Hook Length Theorem states that the number of SYTs is $$ (Error compiling LaTeX. Unknown error_msg)\frac{