Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 StandardYoungTableaux

Source Notebook

Enumerate all standard Young tableaux of a given shape

Contributed by: Jan Mangaldan

ResourceFunction["StandardYoungTableaux"][p]

generates a list of all standard Young tableaux with shape corresponding to the integer partition p.

Details

A standard Young tableau is a Young tableau in which the numbers form an increasing sequence along each line and along each column.

Examples

Basic Examples (1) 

The standard Young tableaux corresponding to the integer partition (2,1,1):

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

Scope (2) 

Visualize all the standard Young tableaux with shape (3,2,1,1):

In[2]:=
Grid[Map[Function[x, Item[x, Frame -> True]], #, {2}]] & /@ ResourceFunction["StandardYoungTableaux"][{3, 2, 1, 1}]
Out[2]=

Count the standard Young tableaux:

In[3]:=
Length[%]
Out[3]=

Properties and Relations (3) 

All the results of StandardYoungTableaux satisfy the resource function TableauQ:

In[4]:=
yt = ResourceFunction["StandardYoungTableaux"][{4, 3}]
Out[4]=
In[5]:=
ResourceFunction["TableauQ"] /@ yt
Out[5]=

Count the number of standard Young tableaux of a given shape:

In[6]:=
yt = ResourceFunction["StandardYoungTableaux"][{3, 2}]
Out[6]=
In[7]:=
Length[%]
Out[7]=

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

In[8]:=
ResourceFunction["NumberOfTableaux"][{3, 2}]
Out[8]=

Count all the standard Young tableaux corresponding to all partitions of a given number:

In[9]:=
Flatten[ResourceFunction["StandardYoungTableaux"] /@ IntegerPartitions[5], 1]
Out[9]=
In[10]:=
Length[%]
Out[10]=

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

In[11]:=
ResourceFunction["NumberOfTableaux"][5]
Out[11]=

Version History

  • 1.0.0 – 07 February 2022

Related Resources

Author Notes

Parts of the implementation are adapted from the GroupMath package by Renato Fonseca (https://renatofonseca.net/groupmath).

License Information