Number of Partitions as Coefficient of Power Series

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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


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Historical Note

The expression for the Number of Partitions as Coefficient of Power Series was first noticed by Leonhard Paul Euler.


Sources

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