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ConjugatePartition (1.0.0) current version: 1.1.0 »

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Switch the rows and columns of a partition

Contributed by: Wolfram Staff

ResourceFunction["ConjugatePartition"][p]

gives the partition that switches the rows and columns of the integer partition p.

Details and Options

A partition of n is a list of weakly decreasing positive integers that add up to n. For instance, written compactly, these are the five partitions of 4: 4, 31, 22, 211, 1111.

Examples

Basic Examples

Here is a partition of 10 and its Ferrers diagram:

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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" -> CloudObject[ "https://www.wolframcloud.com/obj/bfd1afa3-668c-4583-9ee4-\ 516df50e49c8"], "ShortName" -> "FerrersDiagram"], { ResourceSystemBase -> Automatic}]][p]$

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Here is the conjugate partition and its Ferrers diagram:

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In[4]:=

$ResourceFunction[ ResourceObject[ Association[ "Name" -> "FerrersDiagram", "UUID" -> "ff9f6ee9-e173-4868-997b-754fe2a50d9a", "ResourceType" -> "Function", "Version" -> "1.0.0", "Description" -> "Displays the Ferrers diagram of a partition \ with dots", "RepositoryLocation" -> URL[ "https://www.wolframcloud.com/objects/resourcesystem/api/1.0"], "WolframLanguageVersionRequired" -> "10.0", "SymbolName" -> "FunctionRepository`$\ cfb969414cd645eaa8530fdec3494798`FerrersDiagram", "FunctionLocation" -> CloudObject[ "https://www.wolframcloud.com/obj/bfd1afa3-668c-4583-9ee4-\ 516df50e49c8"], "ShortName" -> "FerrersDiagram"], { ResourceSystemBase -> Automatic}]][q]$

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Some partitions are self-conjugate:

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A self-conjugate partition has a symmetric Ferrers diagram:

In[7]:=

$ResourceFunction[ ResourceObject[ Association[ "Name" -> "FerrersDiagram", "UUID" -> "ff9f6ee9-e173-4868-997b-754fe2a50d9a", "ResourceType" -> "Function", "Version" -> "1.0.0", "Description" -> "Displays the Ferrers diagram of a partition \ with dots", "RepositoryLocation" -> URL[ "https://www.wolframcloud.com/objects/resourcesystem/api/1.0"], "WolframLanguageVersionRequired" -> "10.0", "SymbolName" -> "FunctionRepository`$\ cfb969414cd645eaa8530fdec3494798`FerrersDiagram", "FunctionLocation" -> CloudObject[ "https://www.wolframcloud.com/obj/bfd1afa3-668c-4583-9ee4-\ 516df50e49c8"], "ShortName" -> "FerrersDiagram"], { ResourceSystemBase -> Automatic}]][s]$

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Related Links

Partition–Wolfram MathWorld

Version History

1.1.0 – 21 February 2024
1.0.1 – 31 January 2022
1.0.0 – 31 May 2019

Related Resources

License Information

This work is licensed under a Creative Commons Attribution 4.0 International License