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PartitionsP

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

PartitionsP[n]
    gives the number p(n) of unrestricted partitions of the integer n.
n must be an integer; p(n) = 0 for n < 0. Threads over lists.
For the partitions themselves use IntegerPartitions[n].

Examples

All examples below are verified against the current Mathilda build.

In[1]:= Table[PartitionsP[k], {k, 0, 12}]
Out[1]= {1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77}

In[2]:= PartitionsP[100]
Out[2]= 190569292

In[3]:= PartitionsP[4096]
Out[3]= 6927233917602120527467409170319882882996950147283323368445315320451

In[4]:= Table[Times @@ PartitionsP[Last /@ FactorInteger[n]], {n, 12}]
Out[4]= {1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2}

Implementation notes

  • Protected, Listable — PartitionsP[{2, 4, 6}] → {2, 5, 11}.
  • Two engines, dispatched by the size of n (threshold n = 1000):
  • Small n — Euler's pentagonal-number-theorem recurrence

Attributes: Listable, Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References