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


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


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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 nonisomorphic 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:

Introduced in 1988