A partition is a way of writing an integer as a sum of positive integers where the order of the addends is not significant, possibly subject to one or more additional constraints. By convention, partitions are normally written from largest to smallest addends (Skiena 1990, p. 51), for example, . All the partitions of a given positive integer can be generated in the Wolfram Language using IntegerPartitions[list]. PartitionQ[p] in the Wolfram Language package Combinatorica` can be used to test if a list consists of positive integers and therefore is a valid partition.
Andrews (1998, p. 1) uses the notation to indicate "a sequence is a partition of ," and the notation , known as the frequency representation, to abbreviate the partition .
The partitions on a number correspond to the set of solutions to the Diophantine equation
For example, the partitions of four, given by (1, 1, 1, 1), (1, 1, 2), (2, 2), (4), and (1, 3) correspond to the solutions , (2, 1, 0, 0), (0, 2, 0, 0), (0, 0, 0, 1), and (1, 0, 1, 0).
Particular types of partition functions include the partition function P, giving the number of partitions of a number as a sum of smaller integers without regard to order, and partition function Q, giving the number of ways of writing the integer as a sum of positive integers without regard to order and with the constraint that all integers in each sum are distinct. The partition function bk, which gives the number of partitions of in which no parts are multiples of , is sometimes also used (Gordon and Ono 1997).
The Euler transform gives the number of partitions of into integer parts of which there are different types of parts of size 1, of size 2, etc. For example, if for all , then is the number of partitions of into integer parts. Similarly, if for prime and for composite, then is the number of partitions of into prime parts (Sloane and Plouffe 1995, p. 21).
A partition of a number into a sum of elements of a list can be determined using a greedy algorithm. The following table gives the number of partitions of into a sum of positive powers for multiples of .