# nLab partition function (number theory)

Contents

This entry is about the notion in number theory/combinatorics. For partition functions in the sense of statistical mechanics and quantum field theory see at partition function.

### Context

#### Combinatorics

combinatorics

enumerative combinatorics

graph theory

rewriting

### Polytopes

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category: combinatorics

# Contents

## Idea

In number theory/combinatorics, the function which assigns to a natural number $n \in \mathbb{N}$ the number $p(n)$ of its partitions (into multisets of positive natural numbers) is often called the paritition function.

## Properties

### Generating function

Its (ordinary) generating function is

$\sum_{n=0}^\infty p(n) x^n = \prod_{k=1}^\infty (1 - x^k)^{-1} \,.$

### Asymptotic expansion

An asymptotic expansion of the partition function is (Hardy & Ramanujan 1918, see DeSalvo 20 for discussion)

$p(n) \;\sim\; \frac { e^{ \sqrt{2/3}\pi \cdot \sqrt{n} } } { 4 \sqrt{3} \cdot n} \,.$