Partitions and q-Series

Integer partitions represent one of the deepest subjects in combinatorics, with connections to number theory, representation theory, and mathematical physics. Generating functions for partitions, expressed as q-series, reveal remarkable algebraic structures.

DefinitionInteger Partition

A partition of a positive integer $n$ is a way of writing $n$ as a sum of positive integers, where the order doesn't matter. We write partitions in non-increasing order: $n = \lambda_1 + \lambda_2 + \cdots + \lambda_k$ where $\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_k > 0$ .

The number of partitions of $n$ is denoted $p(n)$ , with $p(0) = 1$ by convention. For example:

$p(4) = 5$ : the partitions are $4, 3+1, 2+2, 2+1+1, 1+1+1+1$
$p(5) = 7$ : partitions are $5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1$

DefinitionPartition Generating Function

The generating function for $p(n)$ is: $P(q) = \sum_{n=0}^\infty p(n) q^n = \prod_{k=1}^\infty \frac{1}{1-q^k}$

This product formula (Euler's partition identity) states that each factor $\frac{1}{1-q^k} = 1 + q^k + q^{2k} + q^{3k} + \cdots$ corresponds to using the part $k$ zero, one, two, or more times in a partition.

The coefficient of $q^n$ in the infinite product counts partitions of $n$ by independently choosing how many times each positive integer appears as a part.

ExampleRestricted Partitions

Different restrictions on partitions lead to different q-series:

Partitions into distinct parts: $\prod_{k=1}^\infty (1 + q^k)$
Partitions into odd parts: $\prod_{k=1}^\infty \frac{1}{1-q^{2k-1}}$
Partitions with at most $m$ parts: $\prod_{k=1}^m \frac{1}{1-q^k}$
Partitions with parts at most $m$ : $\prod_{k=1}^m \frac{1}{1-q^k}$ (same as above!)
Self-conjugate partitions (equal to their Ferrers diagram transpose): Count equals partitions into distinct odd parts

TheoremEuler's Partition Theorem

The number of partitions of $n$ into distinct parts equals the number of partitions of $n$ into odd parts. Algebraically: $\prod_{k=1}^\infty (1 + q^k) = \prod_{k=1}^\infty \frac{1}{1 - q^{2k-1}}$

Remark

This theorem has a beautiful bijective proof using the binary representation of parts. Each partition into odd parts can be uniquely written by grouping equal parts and using the binary expansion of their multiplicities.

DefinitionFerrers Diagram and Young Tableaux

A Ferrers diagram (or Young diagram) is a visual representation of a partition using rows of dots or boxes, arranged left-justified with rows in non-increasing order. For partition $(5,3,3,1)$ :

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The conjugate partition $\lambda'$ is obtained by reflecting the Ferrers diagram across the main diagonal, swapping rows and columns.

ExamplePentagonal Number Theorem

Euler's pentagonal number theorem is one of the most beautiful identities in partition theory: $\prod_{k=1}^\infty (1 - q^k) = \sum_{n=-\infty}^\infty (-1)^n q^{n(3n-1)/2}$

The exponents $\frac{n(3n-1)}{2}$ are the pentagonal numbers: $0, 1, 2, 5, 7, 12, 15, 22, \ldots$ (for $n = 0, \pm 1, \pm 2, \ldots$ ).

This gives a recurrence for $p(n)$ : $p(n) = p(n-1) + p(n-2) - p(n-5) - p(n-7) + p(n-12) + \cdots$ with signs alternating in pairs and pentagonal number offsets.

Remark

Partition theory connects deeply to:

Modular forms: The generating function $P(q)$ is related to Dedekind's eta function $\eta(\tau) = q^{1/24}\prod_{n=1}^\infty (1-q^n)$
Representation theory: Partitions index irreducible representations of symmetric groups
q-analogs: Many classical formulas have q-analogs expressed through partition generating functions
Physics: Partition functions in statistical mechanics and string theory

Ramanujan discovered numerous deep congruences, such as $p(5n+4) \equiv 0 \pmod{5}$ , revolutionizing the field.

The theory of partitions and q-series remains one of the most active areas of research in modern combinatorics.