A partition is a way of writing a nonnegative integer 
as a sum of positive integers
where the order of the addends is not significant, possibly
subject to one or more additional constraints. The empty partition is the unique
partition of 0. 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[n].
Equivalently, in the terminology of the twelvefold way, the partitions of 
into at most 
parts count the ways of distributing 
identical balls into 
identical boxes.
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 
.
For a positive integer 
,
its partitions correspond to the set of solutions 
in nonnegative
integers 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 
as a sum of positive
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 
otherwise, then 
is the number of partitions of 
into prime parts (Sloane and
Plouffe 1995, p. 21).
A greedy algorithm that repeatedly chooses the largest allowed part need not find a partition, even when one exists. For example,
with 
and the list 
, choosing 4 leaves an unrepresentable remainder
of 2, although 
is a valid partition. Partitions with parts in 
can be enumerated in the Wolfram
Language using IntegerPartitions[n,
All, L]. The following table gives the number of partitions of 
into positive integer 
th powers.
 |  |  |  |  |
| Sloane | A000041 | A001156 | A003108 | A046042 |
| 10 | 42 | 4 | 2 | 1 |
| 50 | 204226 | 104 | 10 | 4 |
| 100 | 190569292 | 1116 | 39 | 9 |
| 150 | 40853235313 | 6521 | 97 | 15 |
| 200 | 3972999029388 | 27482 | 208 | 24 |
| 250 | 230793554364681 | 94987 | 388 | 34 |
| 300 | 9253082936723602 | 284316 | 683 | 49 |
Partitions can also be counted by their number of parts. If 
is the number of partitions of 
into exactly 
parts, then 
is a unimodal
sequence for every 
. Padhi (2026) proved this
Auluck-Chowla-Gupta conjecture,
completing the previously asymptotic result. Padhi (2026) reports Claude assistance
with computations and code.
Another statistic of partitions is the smallest parts function, which counts the occurrences of their smallest parts with multiplicity. Rolen (2026) classified its canonical prime congruences, showing that the only primes are 5, 7, and 13.
Biases between parts in different arithmetic progressions have an explicit limiting law. Let 
be positive integers
with 
and 
,
and put 
and 
. For a partition chosen uniformly from
all partitions of 
, let 
count its parts in 
, with multiplicity.
Li and Zhao (2026) proved
 |
where 
is the regularized
beta function. The specified starting points of the arithmetic
progressions matter.
