Partition function

The integer partition function p(n) counts the ways a positive integer n can be expressed as a sum of positive integers, disregarding order.

Integer partition: An unordered expression of a nonnegative integer as a sum of positive integers. Each value p(n) counts the integer partitions of n.
Euler's recurrence for the partition function: A recurrence expressing p(n) through earlier partition numbers using alternating signs and generalized pentagonal numbers. It computes each value from smaller arguments.
Leonhard Euler: An eighteenth-century Swiss mathematician who made foundational contributions across mathematics and physics. His 1740s work established the generating product for partition numbers.
Restricted partition function: A function counting partitions subject to constraints on allowed parts or their multiplicities. It refines p(n) by imposing conditions on the summands.
Parity of the partition function: The study of which partition numbers p(n) are even or odd. No simple complete pattern for the parity of p(n) is known.
Partition number: The number of integer partitions of a specified nonnegative integer. p(n) is the partition number assigned to n.
Hardy–Ramanujan asymptotic formula: An asymptotic estimate for p(n), led by exp(π√(2n/3))/(4n√3). It describes the rapid growth of p(n) for large n.
Srinivasa Ramanujan: An Indian mathematician whose work in number theory produced influential results on partitions and modular forms. His congruences and formulas reshaped the study of p(n).
Partition identity: An equality between counts of two classes of integer partitions. Generating functions and bijections prove identities between partition counts.
Partition function modulo powers of 2: The study of congruences for p(n) modulo 2ᵏ as n varies. Higher powers of two reveal unresolved distribution questions beyond parity.

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