PartitionsP [n]
gives the number p(n) of unrestricted partitions of the integer n.
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
open all close allBasic 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 :
See Also
PartitionsQ DedekindEta IntegerPartitions FiniteAbelianGroupCount
Function Repository: PartitionRank PartitionCrank
Related Links
History
Introduced in 1988 (1.0)
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: 16-November-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: 16-November-2025]}