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PartitionsP






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Integer Functions > PartitionsP[n] > Introduction to partitions





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 .





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