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PartitionsP[n]

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

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Basic Examples  
Scope  
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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)

Compute the first few values:

Wolfram Language code: Table[PartitionsP[k], {k, 0, 12}]

Plot the number of unrestricted partitions:

Wolfram Language code: DiscretePlot[Log[10, PartitionsP[k]], {k, 100}]

Scope  (3)

Compute the number of partitions for large numbers:

Wolfram Language code: Table[PartitionsP[2 ^ k], {k, 0, 12}]

PartitionsP threads element-wise over lists:

Wolfram Language code: PartitionsP[{2, 4, 6}]

TraditionalForm formatting:

Wolfram Language code: PartitionsP[n]//TraditionalForm

Applications  (3)

Number of non‐isomorphic Abelian groups of order n:

Wolfram Language code: Table[Times@@PartitionsP[Last /@ FactorInteger[n]], {n, 12}]

Compare to FiniteAbelianGroupCount:

Wolfram Language code: Table[FiniteAbelianGroupCount[n], {n, 12}]

Compare cumulative counts of even and odd partitions:

Wolfram Language code: ListPlot[Accumulate[Table[Mod[PartitionsP[k], 2], {k, 1000}]] / Range[1000]]

Visualize p-adic valuations of the number of partitions:

Wolfram Language code: ArrayPlot[Mod[Table[IntegerExponent[Table[PartitionsP[n], {n, 120}], k], {k, 2, 120}], 2]]

Properties & Relations  (4)

PartitionsP gives the length of IntegerPartitions:

Wolfram Language code: IntegerPartitions[5]
Wolfram Language code: PartitionsP[5]

Obtain values of PartitionsP from series expansion:

Wolfram Language code: Series[1 / QPochhammer[t], {t, 0, 20}]

Use FullSimplify to simplify expressions containing PartitionsP:

Wolfram Language code: FullSimplify[Mod[PartitionsP[5n + 4], 5] == 0, Element[n, Integers] && n > 0]

FindSequenceFunction can recognize the PartitionsP sequence:

Wolfram Language code: Table[PartitionsP[n], {n, 10}]
Wolfram Language code: FindSequenceFunction[%, n]

Possible Issues  (1)

PartitionsP evaluates only for integer arguments:

Wolfram Language code: PartitionsP[12.1]

Use Simplify to find implicit integers in arguments:

Wolfram Language code: PartitionsP[12 + (E + 1) ^ 2 - Expand[(E + 1) ^ 2]]
Wolfram Language code: Simplify[%]

Neat Examples  (2)

Successive differences of PartitionsP modulo 2:

Wolfram Language code: ArrayPlot[Mod[NestList[Differences, PartitionsP[Range[100]], 100], 2]]

A "random" walk based on PartitionsP:

Wolfram Language code: ListLinePlot[ReIm /@ Accumulate[Exp[3.4 Pi I PartitionsP[Range[500]]]]]
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_2026_partitionsp, author="Wolfram Research", title="{PartitionsP}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/PartitionsP.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_partitionsp, organization={Wolfram Research}, title={PartitionsP}, year={1988}, url={https://reference.wolfram.com/language/ref/PartitionsP.html}, note=[Accessed: 12-June-2026]}