Function Repository Resource:

HookLengths (2.0.0) current version: 2.0.1 »

Get the list of lists whose entries are the hook lengths of the entries of a Young diagram

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 diagram of p.

Details and Options

In the Young diagram 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

Here is a partition:

In[1]:=

This is its Ferrers diagram:

In[2]:=

$ResourceFunction[ ResourceObject[ Association[ "Name" -> "FerrersDiagram", "UUID" -> "ff9f6ee9-e173-4868-997b-754fe2a50d9a", "ResourceType" -> "Function", "Version" -> "1.0.0", "Description" -> "Displays the Ferrers diagram of a partition \ with dots", "RepositoryLocation" -> URL[ "https://www.wolframcloud.com/objects/resourcesystem/api/1.0"], "WolframLanguageVersionRequired" -> "10.0", "SymbolName" -> "FunctionRepository`$\ cfb969414cd645eaa8530fdec3494798`FerrersDiagram", "FunctionLocation" -> LocalObject[ "file:///Users/georgebeck/Library/Wolfram/Objects/Resources/ff9/\ ff9f6ee9-e173-4868-997b-754fe2a50d9a/download/DefinitionData"], "ShortName" -> "FerrersDiagram"], ResourceSystemBase -> "https://www.wolframcloud.com/objects/\ resourcesystem/api/1.0"]]@p$

Out[2]=

These are the hook lengths for the shape p:

In[3]:=

Out[3]=

This is used to format a Young diagram:

In[4]:=

$formatyoungdiagram[p_] := Module[ {mat, mmax, nmax}, mat = PadRight[Table[1, #] & /@ p]; {mmax, nmax} = Dimensions@mat; Frame -> {None, None, Select[Flatten[ Table[{m, n} -> mat[[m, n]] == 1, {m, mmax}, {n, nmax}]], True == Last@# &]} ]$