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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

Arithmetic

Contents

Idea

In number theory/combinatorics, the function which assigns to a natural number nn \in \mathbb{N} the number p(n)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

n=0 p(n)x n= k=1 (1x k) 1. \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)e 2/3πn43n. p(n) \;\sim\; \frac { e^{ \sqrt{2/3}\pi \cdot \sqrt{n} } } { 4 \sqrt{3} \cdot n} \,.

References

See also:

Asymptotic behaviour:

Created on June 3, 2021 at 06:44:05. See the history of this page for a list of all contributions to it.