html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 PartitionsQ

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 .

© 1998-2013 Wolfram Research, Inc.