Function Repository Resource:

HookLengths

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):

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Visualize the corresponding Ferrers diagram:

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Scope (4)

Here is a partition:

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The Young diagram with shape p:

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A hook of length 5:

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The hook lengths of p:

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Properties and Relations (3)

Here is a partition:

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

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

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George Beck

Version History

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