|
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)  .
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  .
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  .
|
