Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 NumberOfTableaux

Source Notebook

Get a count of Young tableaux for a given size or shape

Contributed by: Ed Pegg Jr and Steven Skiena

ResourceFunction["NumberOfTableaux"][t]

returns the number of Young tableaux with shape t, where the shape is a decreasing integer partition.

ResourceFunction["NumberOfTableaux"][n]

returns the total number of Young tableaux for all decreasing integer partitions of n.

Details

A Young tableau is a list of lists of decreasing length partitioning a permutation. For all entries in a Young tableau, t[[a,b]]<t[[a+1,b]] and t[[a,b]]<t[[a,b+1]].
Young tableaux are often called tableaux.
A few pictures for items related to Young tableaux:
Young diagram of Young tableau {{1,2,4},{3,5}}, English notation
Young diagram of Young tableau {{1,2,4},{3,5}}, French notation
Ferrers diagrams of Young tableau {{1,2,4},{3,5}}
{{1,2,4},{3,5}}Young tableau with shape {3,2}

Examples

Basic Examples (1) 

Count the tableaux with shape {3,2}:

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

Scope (2) 

Find the integer partitions of 6:

In[2]:=
ip = IntegerPartitions[6]
Out[2]=

Using the partitions as shapes, find the tableaux count for each shape:

In[3]:=
ResourceFunction["NumberOfTableaux"] /@ ip
Out[3]=

Calculate the total:

In[4]:=
Total[%]
Out[4]=

Find the number of tableaux with order 6:

In[5]:=
ResourceFunction["NumberOfTableaux"][6]
Out[5]=

Determine the shape of an order-15 tableau:

In[6]:=
Length /@ {{1, 4, 5, 7, 10, 13}, {2, 9, 15}, {3, 12}, {6, 14}, {8}, {11}}
Out[6]=

Count tableaux with that shape:

In[7]:=
ResourceFunction["NumberOfTableaux"][{6, 3, 2, 2, 1, 1}]
Out[7]=

Version History

  • 1.0.0 – 24 January 2022

Related Resources

License Information