Integer partitions

Integer partitions refer to the ways of expressing a positive integer as the sum of one or more positive integers. The order of terms in each sum does not matter; for example, the two sums \(4 = 1 + 1 + 1 + 1\) and \(4 = 2 + 2\) represent two distinct partitions of the integer 4.

In the context of combinatorial mathematics, especially in the theory of partitions, the "crank" is a statistic associated with partitions of integers. It was introduced by the mathematician George Andrews and has applications in the study of partition theory and modular forms. A partition of a positive integer is a way of writing it as a sum of positive integers, where the order of addends does not matter.

Glaisher's theorem is a result in number theory, specifically related to the distribution of prime numbers. It gives a bound on the error term in the prime number theorem. The prime number theorem states that the number of primes less than a given number \( x \) is asymptotically equivalent to \( \frac{x}{\log x} \). Glaisher's theorem refines the understanding of the error in this approximation.

In number theory, a partition of a positive integer \( n \) is a way of writing \( n \) as the sum of positive integers, where the order of the summands does not matter. For example, the integer \( 4 \) can be partitioned into the following distinct sums: 1. \( 4 \) 2. \( 3 + 1 \) 3. \( 2 + 2 \) 4.

The Pentagonal Number Theorem is a result in number theory associated with the generating function for partition numbers. Specifically, it relates to the representation of integers as sums of pentagonal numbers.

The term "rank of a partition" can refer to different concepts depending on the context in which it is used, such as in mathematics, particularly in number theory and combinatorics, or in the study of partitions in linear algebra (like matrix ranks or partitions of sets). In the context of number theory and partitions, the rank of a partition refers to the number of parts (or summands) in the partition minus the largest part.