Number of Partitions as Coefficient of Power Series

Theorem

The number of partitions $\map p n$ of a (strictly) positive integer $n$ is equal to the coefficient of $x^n$ when the expression:

$\map f n = \dfrac 1 {\paren {1 - x} \paren {1 - x^2} \paren {1 - x^3} \cdots}$

is expanded into a power series.

That is:

$\map f n = 1 + \map p 1 x + \map p 2 x^2 + \map p 3 x^3 + \cdots$

or:

$\ds \sum_{n \mathop \in \Z_{\ge 0} } \map p n q^n = \prod_{j \mathop \in \Z_{>0} } \dfrac 1 {1 - q^j}$

where $\map p 0 := 1$.

Proof

$\ds \map f n$

$=$

$\ds \dfrac 1 {\paren {1 - x} \paren {1 - x^2} \paren {1 - x^3} \paren {1 - x^4} \cdots}$

$\ds $

$=$

$\ds \frac 1 {\paren {1 - x^1} } \times \frac 1 {\paren {1 - x^2} } \times \frac 1 {\paren {1 - x^3} } \times \frac 1 {\paren {1 - x^4} } \times \cdots$

$\ds $

$=$

$\ds \paren {1 + x^1 + x^{1 + 1} + x^{1 + 1 + 1} + \cdots} \times \paren {1 + x^2 + x^{2 + 2} + x^{2 + 2 + 2} + \cdots} \times \paren {1 + x^3 + x^{3 + 3} + x^{3 + 3 + 3} + \cdots} \times \paren {1 + x^4 + x^{4 + 4} + x^{4 + 4 + 4} + \cdots} \times \cdots$

Sum of Infinite Geometric Sequence

$\ds $

$=$

$\ds 1 + \paren {x^1 } + \paren {x^{1 + 1} + x^2 } + \paren {x^{1 + 1 + 1} + x^1 x^2 + x^3 } + \paren {x^{1 + 1 + 1 + 1} + x^{1 + 1} x^2 + x^{2 + 2} + x^1 x^3 + x^4} + \cdots$

$\ds $

$=$

$\ds 1 + \paren {x^1 } + \paren {x^{1 + 1} + x^2 } + \paren {x^{1 + 1 + 1} + x^{1 + 2} + x^3 } + \paren {x^{1 + 1 + 1 + 1} + x^{1 + 1 + 2} + x^{2 + 2} + x^{1 + 3} + x^4} + \cdots$

Product of Powers

$\ds $

$=$

$\ds 1 + \map p 1 x + \map p 2 x^2 + \map p 3 x^3 + \map p 4 x^4 + \cdots$

$\ds $

$=$

$\ds \sum_{n \mathop = 0}^\infty \map p n x^n$

$\blacksquare$

Historical Note

The expression for the was first noticed by Leonhard Paul Euler.

Sources

1954: G. Polya: Induction and Analogy in Mathematics: Chapter $6$
1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.21$: Euler ($\text {1707}$ – $\text {1783}$)
1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.9$: Generating Functions: Exercise $8$
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): partition function