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
Related topics
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.