Definitions of partitions

The partition functions discussed here include two basic functions that describe the structure of integer numbers—the number of unrestricted partitions of an integer (partitions P) , and the number of partitions of an integer into distinct parts (partitions Q) .

Partitions P

For nonnegative integer , the function is the number of unrestricted partitions of the positive integer into a sum of strictly positive numbers that add up to independent of the order, when repetitions are allowed.

The function can be described by the following formulas:

where (with ) is the coefficient of the term in the series expansion around of the function : .

Example: There are seven possible ways to express 5 as a sum of nonnegative integers: . For this reason .

Partitions Q

For nonnegative integer , the function is the number of restricted partitions of the positive integer into a sum of distinct positive numbers that add up to when order does not matter and repetitions are not allowed.

The function can be described by the following formulas:

where (with ) is the coefficient of the term in the series expansion around of the function : .

Example: There are three possible ways to express 5 as a sum of nonnegative integers without repetitions: . For this reason .