Function Repository Resource:

# HookLengths

Get the list of lists whose entries are the hook lengths of the entries of a Young tableau

Contributed by: George Beck
 ResourceFunction["HookLengths"][p] gives a list of lists of lengths p for a partition p where an entry is the hook length of the corresponding cell in the Young tableau of p.

## Details

In the Young tableau of shape p, the hook for a cell c is the set of cells containing c, the cells to the right of c and the cells below c.
The hook length of a cell c is the number of cells in the hook of c.

## Examples

### Basic Examples (2)

The hook lengths of the integer partition (3,2,1,1):

 In:= Out= Visualize the corresponding Ferrers diagram:

 In:= Out= ### Scope (4)

Here is a partition:

 In:= The Young diagram with shape p:

 In:= Out= A hook of length 5:

 In:= Out= The hook lengths of p:

 In:= Out= ### Properties and Relations (3)

Here is a partition:

 In:= The number of standard Young tableaux for the diagram of shape p (the hook length formula):

 In:= Out= This is the same as the result of the resource function NumberOfTableaux:

 In:= Out= George Beck

## Version History

• 2.0.1 – 07 February 2022
• 2.0.0 – 10 July 2019
• 1.0.0 – 18 February 2019