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PartitionsP[n]

gives the number p(n) of unrestricted partitions of the integer n.

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Examples  
Basic Examples  
Scope  
Applications  
Properties & Relations  
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PartitionsP

PartitionsP[n]

gives the number p(n) of unrestricted partitions of the integer n.

Details

  • Integer mathematical function, suitable for both symbolic and numerical manipulation.
  • PartitionsP automatically threads over lists.

Examples

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Basic Examples  (2)

Plot the number of unrestricted partitions:

Scope  (3)

Compute the number of partitions for large numbers:

PartitionsP threads element-wise over lists:

TraditionalForm formatting:

Applications  (3)

Number of non‐isomorphic Abelian groups of order n:

Compare to FiniteAbelianGroupCount:

Compare cumulative counts of even and odd partitions:

Visualize p-adic valuations of the number of partitions:

Properties & Relations  (4)

PartitionsP gives the length of IntegerPartitions:

Obtain values of PartitionsP from series expansion:

Use FullSimplify to simplify expressions containing PartitionsP:

FindSequenceFunction can recognize the PartitionsP sequence:

Possible Issues  (1)

PartitionsP evaluates only for integer arguments:

Use Simplify to find implicit integers in arguments:

Neat Examples  (2)

Successive differences of PartitionsP modulo 2:

A "random" walk based on PartitionsP:

Wolfram Research (1988), PartitionsP, Wolfram Language function, https://reference.wolfram.com/language/ref/PartitionsP.html.

Text

Wolfram Research (1988), PartitionsP, Wolfram Language function, https://reference.wolfram.com/language/ref/PartitionsP.html.

CMS

Wolfram Language. 1988. "PartitionsP." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PartitionsP.html.

APA

Wolfram Language. (1988). PartitionsP. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PartitionsP.html

BibTeX

@misc{reference.wolfram_2025_partitionsp, author="Wolfram Research", title="{PartitionsP}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/PartitionsP.html}", note=[Accessed: 06-August-2025]}

BibLaTeX

@online{reference.wolfram_2025_partitionsp, organization={Wolfram Research}, title={PartitionsP}, year={1988}, url={https://reference.wolfram.com/language/ref/PartitionsP.html}, note=[Accessed: 06-August-2025]}