Integer Partitions

An integer partition expresses a positive integer as an unordered sum of positive integers, and the theory counts and relates such representations.

Definition

A partition of a positive integer n is a way of writing n as a sum of positive integers in which order does not matter; the partition function p(n) counts the number of such partitions.

Scope

This topic studies the partition function p(n), partition identities, Young diagrams and conjugation, and the generating-function methods that encode partitions as infinite products. It includes classical results such as Euler's identity equating partitions into distinct parts with partitions into odd parts, and the Rogers-Ramanujan identities, linking combinatorics to number theory and q-series.

Core questions

• How many ways can a positive integer be written as an unordered sum of positive integers?
• What generating function encodes the partition numbers?
• Which restrictions on parts yield equinumerous partition families?
• How does p(n) grow asymptotically?

Key concepts

• Partition function p(n)
• Young (Ferrers) diagrams
• Conjugate partition
• Distinct and odd parts
• Pentagonal number theorem
• Rogers-Ramanujan identities

Key theories

Euler's product generating function

The generating function for p(n) is the infinite product of 1/(1-x^k) over all positive integers k; manipulating this product yields partition identities and recurrences such as Euler's pentagonal number theorem.

Euler's distinct-odd identity

The number of partitions of n into distinct parts equals the number into odd parts, a foundational partition identity provable bijectively or by a simple generating-function argument.

Clinical relevance

Partition theory connects to representation theory of the symmetric group (where partitions index irreducible representations), statistical mechanics (where partition-like sums appear in lattice models), and the study of modular forms in number theory.

History

Euler launched the modern theory of partitions through generating functions in the 18th century; Hardy and Ramanujan's 1918 asymptotic formula for p(n) inaugurated the analytic study of partition growth.

Key figures

• Leonhard Euler
• Srinivasa Ramanujan
• Godfrey Harold Hardy

Seminal works

• stanley2011
• flajolet2009

Frequently asked questions

How does a partition differ from a composition?

A composition counts ordered sums, so 2+1 and 1+2 are different, whereas a partition treats them as the same since order is ignored.

What is a Young diagram?

A Young diagram represents a partition as left-justified rows of cells, one row per part, giving a visual tool for proving identities by reflecting the diagram.