Function Repository Resource:

# HookLengths

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

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 diagram of p.

## Details and Options

In the Young diagram 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

Here is a partition:

 In[1]:=

This is its Ferrers diagram:

 In[2]:=
 Out[2]=

These are the hook lengths for the shape p:

 In[3]:=
 Out[3]=

This is used to format a Young diagram:

 In[4]:=

The Young diagram with shape p:

 In[5]:=
 Out[5]=

A hook of length 5:

 In[6]:=
 Out[6]=

The hook lengths of p:

 In[7]:=
 Out[7]=

### Applications

The number of standard Young tableaux for the diagram of shape p:

 In[8]:=
 Out[8]=