Function Repository Resource:

HookLengths

Source Notebook

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[1]:=
ResourceFunction["HookLengths"][{3, 2, 1, 1}]
Out[1]=

Visualize the corresponding Ferrers diagram:

In[2]:=
ResourceFunction["FerrersDiagram"][{3, 2, 1, 1}]
Out[2]=

Scope (4) 

Here is a partition:

In[3]:=
p = {5, 3, 1, 1};

The Young diagram with shape p:

In[4]:=
Grid[Map[Item[#, Frame -> True] &, Table[" ", #] & /@ p, {2}]]
Out[4]=

A hook of length 5:

In[5]:=
Grid[Map[Item[#, Frame -> True] &, {{" ", "x", "x", "x", "x"}, {" ", "x", " "}, {" "}, {" "}}, {2}]]
Out[5]=

The hook lengths of p:

In[6]:=
Grid[Map[Item[#, Frame -> True] &, ResourceFunction["HookLengths"][p], {2}]]
Out[6]=

Properties and Relations (3) 

Here is a partition:

In[7]:=
p = {5, 4, 3, 3};

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

In[8]:=
Total[p]!/Apply[Times, ResourceFunction["HookLengths"][p], {0, 1}]
Out[8]=

This is the same as the result of the resource function NumberOfTableaux:

In[9]:=
ResourceFunction["NumberOfTableaux"][p]
Out[9]=

Publisher

George Beck

Version History

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

Related Resources

License Information