Partitionstheorie

Partitionstheorie, or partition theory, is a branch of number theory and combinatorics that studies integer partitions: representations of a positive integer n as a sum of positive integers, where order is irrelevant. The function p(n) denotes the number of partitions of n, and many questions ask for exact values, asymptotics, or identities among partitions. Partitions are often represented by Ferrers diagrams or Young diagrams.

The generating function for p(n) is given by the infinite product sum_{n≥0} p(n) q^n = ∏_{k≥1} (1 − q^k)^{-1}. This product identity is closely tied to Euler’s pentagonal number theorem, which yields a recurrence for p(n): p(n) = p(n−1) + p(n−2) − p(n−5) − p(n−7) + p(n−12) + ... using generalized pentagonal numbers g_k = k(3k−1)/2.

Key results include Euler’s theorem: the number of partitions of n into distinct parts equals the number of partitions into odd parts. The Rogers–Ramanujan identities reveal deep q-series identities relating restricted partitions to infinite products. Hardy and Ramanujan established the asymptotic formula p(n) ~ (1/(4n√3)) exp(π√(2n/3)), later refined by Rademacher into a convergent series.

Modular forms and q-series play a central role: the generating function ∑ p(n) q^n equals 1/η(τ), where η is the Dedekind eta function, linking partition theory to modular forms and to Ramanujan congruences p(5k+4) ≡ 0 mod 5, p(7k+5) ≡ 0 mod 7, p(11k+6) ≡ 0 mod 11.

Variants of partitions include partitions with restricted parts, partitions into exactly k parts, plane partitions, and colored partitions. Partition theory interfaces with representation theory, combinatorics, and mathematical physics, and provides both exact results and powerful asymptotic methods.