Editing Integer partition (section)

{{short description|Decomposition of an integer as a sum of positive integers}}
{{about|partitioning an integer|grouping elements of a set|Partition of a set|the partition calculus of sets|Infinitary combinatorics|the problem of partitioning a multiset of integers so that each part has the same sum|Partition problem}}
[[File:Ferrer partitioning diagrams.svg|thumb|right|300px|[[Young diagram#Diagrams|Young diagrams]] associated to the partitions of the positive integers 1 through 8.  They are arranged so that images under the reflection about the main diagonal of the square are conjugate partitions.]]
[[File:Partitions of n with biggest addend k.svg|thumb|right|300px|Partitions of {{mvar|n}} with largest part {{mvar|k}}]]

In [[number theory]] and [[combinatorics]], a '''partition''' of a non-negative [[integer]] {{mvar|n}}, also called an '''integer partition''', is a way of writing {{mvar|n}} as a [[summation|sum]] of [[positive integers]]. Two sums that differ only in the order of their [[summand]]s are considered the same partition. (If order matters, the sum becomes a [[composition (combinatorics)|composition]].)  For example, {{math|4}} can be partitioned in five distinct ways:

:{{math|4}}
:{{math|3 + 1}}
:{{math|2 + 2}}
:{{math|2 + 1 + 1}}
:{{math|1 + 1 + 1 + 1}}

The only partition of zero is the empty sum, having no parts.

The order-dependent composition {{math|1 + 3}} is the same partition as {{math|3 + 1}}, and the two distinct compositions {{math|1 + 2 + 1}} and {{math|1 + 1 + 2}} represent the same partition as {{math|2 + 1 + 1}}.

An individual summand in a partition is called a '''part'''.  The number of partitions of {{mvar|n}} is given by the [[Partition function (number theory)|partition function]] {{math|''p''(''n'')}}. So {{math|1=''p''(4) = 5}}. The notation {{math|''&lambda;'' ⊢ ''n''}} means that {{mvar|&lambda;}} is a partition of {{mvar|n}}.

Partitions can be graphically visualized with [[Young diagram]]s or [[Ferrers diagram]]s. They occur in a number of branches of [[mathematics]] and [[physics]], including the study of [[symmetric polynomial]]s and of the [[symmetric group]] and in [[group representation|group representation theory]] in general.