Function Repository Resource:

# NumberOfTableaux

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}:

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### Scope (2)

Find the integer partitions of 6:

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

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Calculate the total:

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Find the number of tableaux with order 6:

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Determine the shape of an order-15 tableau:

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Count tableaux with that shape:

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## Version History

• 1.0.0 – 24 January 2022