# hookLengthFormula(Partition) -- a formula for the number of standard tableaux

## Synopsis

• Function: hookLengthFormula
• Usage:
hookLengthFormula(p)
• Inputs:
• p, an instance of the type Partition, a partition that indexes a Specht Module
• Outputs:
• an integer, the dimension of the Specht module S_\lambda

## Description

The hook length formula is a method that counts the number of standard tableaux of a given shape p. Therefore it counts the dimension of the associated Specht module.

For each Ferrer diagram and each cell (a,b) the hook at (a,b) is the set of cells that comprise (a,b) the cells that are below (a,b), and the cells that are to right of (a,b). The hook length of a hook h(a,b) is defined of the number of cells in the hook.

If p is a partition of n then the hook length formula for p is $n!/\prod_{(a,b)} h(a,b)$

 i1 : p = new Partition from {3,2} o1 = Partition{3, 2} o1 : Partition i2 : standardTableaux p o2 = {| 0 1 2 |, | 0 1 3 |, | 0 1 4 |, | 0 2 3 |, | 0 2 4 |} | 3 4 | | 2 4 | | 2 3 | | 1 4 | | 1 3 | o2 : TableauList i3 : hookLengthFormula p o3 = 5