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PartitionsQ[n]

gives the number q(n) of partitions of the integer n into distinct parts.

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Examples  
Basic Examples  
Scope  
Applications  
Properties & Relations  
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Neat Examples  
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PartitionsQ

PartitionsQ[n]

gives the number q(n) of partitions of the integer n into distinct parts.

Details

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

Examples

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Basic Examples  (2)

Compute the first few values:

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

Plot the number of restricted partitions:

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

Scope  (3)

Compute the number of partitions for large numbers:

Wolfram Language code: PartitionsQ[100000]

PartitionsQ threads element-wise over lists:

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

TraditionalForm formatting:

Wolfram Language code: PartitionsQ[n]//TraditionalForm

Applications  (3)

Compare cumulative counts of even and odd partitions into distinct parts:

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

Plot the ratio of the number of partitions with its asymptotic value:

Wolfram Language code: ListPlot[Table[PartitionsQ[k] / ((E^π Sqrt[(k/3)]/4 3^1 / 4 k^3 / 4)), {k, 1000}]]

Visualize p-adic valuations of the number of partitions:

Wolfram Language code: ArrayPlot[Mod[#, 2]&@Table[IntegerExponent[Table[PartitionsQ[n], {n, 120}], k], {k, 2, 120}]]

Properties & Relations  (4)

PartitionsQ gives the length of IntegerPartitions with nonrepeating parts:

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

Generate the explicit partitions:

Wolfram Language code: Select[IntegerPartitions[5], Max[Length /@ Split@ #] == 1&]

Model PartitionsQ based on the definition:

Wolfram Language code: Table[Length[Select[IntegerPartitions[n], Sort[#] == Union[#]&]], {n, 12}]
Wolfram Language code: Table[PartitionsQ[n], {n, 12}]

Obtain values of PartitionsQ from series expansion:

Wolfram Language code: Series[1 / QPochhammer[t, t ^ 2], {t, 0, 20}]

FindSequenceFunction can recognize the PartitionsQ sequence:

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

Possible Issues  (1)

PartitionsQ evaluates only for explicit integers:

Wolfram Language code: PartitionsQ[12.1]

Use Simplify to find implicit integers in arguments:

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

Neat Examples  (2)

Successive differences of PartitionsQ modulo 2:

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

A "random" walk based on PartitionsQ:

Wolfram Language code: ListLinePlot[ReIm /@ Accumulate[Exp[3.4 Pi I PartitionsQ[Range[500]]]]]
Wolfram Research (1988), PartitionsQ, Wolfram Language function, https://reference.wolfram.com/language/ref/PartitionsQ.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_partitionsq, author="Wolfram Research", title="{PartitionsQ}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/PartitionsQ.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_partitionsq, organization={Wolfram Research}, title={PartitionsQ}, year={1988}, url={https://reference.wolfram.com/language/ref/PartitionsQ.html}, note=[Accessed: 12-June-2026]}