Ono's inequality

Ono's inequality is a bound in the theory of integer partitions, associated with Ken Ono's work. It constrains quantities derived from the partition function.

Connect

Partition function: The function p(n) counts the ways to express a positive integer n as a sum of positive integers, disregarding order. Ono's bound concerns quantities built from this central counting function.
Partition inequalities: Partition inequalities compare partition numbers or related quantities using order bounds. Ono's inequality belongs to this broader family of bounds.
Ken Ono: Ken Ono is an American mathematician whose research includes partitions, modular forms, and number theory. The inequality is associated with his work on integer partitions.
Modular form: A modular form is a highly symmetric holomorphic function with specified transformation behavior under modular transformations. Modular forms underlie many modern results about partition numbers.
Integer partition: An integer partition expresses a nonnegative integer as a sum of positive integers, with order disregarded. Partitions are the objects counted by the function constrained in Ono's inequality.
Log-concavity: A positive sequence is log-concave when each squared term is at least the product of its neighbors. Log-concavity is a common shape constraint for sequences and clarifies what a partition bound may assert.
George E. Andrews: George E. Andrews is a mathematician known for research on partitions and q-series. His work established major parts of the modern theory surrounding partition identities and bounds.
Dedekind eta function: The Dedekind eta function is a modular form-related function built from an infinite product in the upper half-plane. Its product encodes the partition generating function up to a simple factor.
Ramanujan congruences: Congruences give divisibility conditions on partition numbers in particular arithmetic progressions. They show another striking kind of structure in the partition numbers Ono studies.
Finite differences: Finite differences measure changes in a sequence through subtraction of neighboring values. Differences of partition numbers are natural candidates for quantitative bounds.