partitietheorie

Partition theory, known as partitietheorie in Dutch, is a branch of number theory and combinatorics that studies partitions of integers. A partition of a positive integer n is a way of writing n as a sum of positive integers, where the order of the summands is irrelevant. For example, the partitions of 4 are 4; 3+1; 2+2; 2+1+1; and 1+1+1+1. The partition function p(n) counts the number of partitions of n, with p(0)=1 by convention.

A central tool is the generating function for partitions: sum_{n≥0} p(n) q^n = ∏_{k≥1} 1/(1 - q^k). This infinite product, discovered by Euler, encodes all partitions and leads to many identities. Euler's pentagonal number theorem expresses ∏_{k≥1} (1 - q^k) as a signed series ∑_{m∈Z} (-1)^m q^{m(3m-1)/2}, which in turn yields recurrence relations for p(n).

Growth and refinements: Hardy and Ramanujan showed that p(n) grows roughly like exp(π√(2n/3)) divided by 4n√3; later, Rademacher produced a convergent refinement. Ramanujan also discovered remarkable congruences for p(n), such as p(5k+4) ≡ 0 (mod 5), p(7k+5) ≡ 0 (mod 7), and p(11k+6) ≡ 0 (mod 11). These results reflect connections between partition theory and modular forms.

Extensions include partitions with restrictions (distinct parts, odd parts), plane partitions, and combinatorial statistics such as rank and crank. Ferrers diagrams visualize partitions, while connections to q-series, symmetric functions, and modular forms place partition theory at the intersection of combinatorics and number theory.