PartitionsP

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

For example, p(5)==7 . There are 7 possibilities to express 5 as a sum of positive integers:
5==1+4==2+3==1+1+3==1+2+2==1+1+1+2==1+1+1+1+1.

Specific values

Values at fixed points

Values at infinities

General characteristics

Domain and analyticity

The partitions p(n) is a nonanalytical function which is defined only for integers.

Symmetries and periodicities

Symmetry

No symmetry

Periodicity

No periodicity

Series representations

Generalized power series

Asymptotic series expansions

Generating functions

Identities

Functional identities

Complex characteristics

Real part

Imaginary part

Absolute value

Argument

Conjugate value

Summation

Finite summation

Infinite summation

Representations through equivalent functions

With related functions

Inequalities

Other identities

Congruence properties

History

– G. W. Leibniz (1669) investigated the number of ways a given positive integer can be decomposed into smaller ones
– L. Euler (1740)
– S. Ramanujan (1917)
– G.H. Hardy (1920) introduced the notation p(n)

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