View source for Partition (combinatorics)

==Definition==
A ''partition'' of  {{M|n}} having {{M|m}} parts<ref name="C">Combinatorics - Russell Merris</ref> is an unordered collection of {{M|m}} integers that sum to {{M|n}}.
* Examples: {{M|6}} may be partitioned into {{M|3}} parts, {{M|3}}, {{M|2}} and {{M|1}}

==Notation==
===Of partitions===
An {{M|m}}-part partition of {{M|n}} is represented by a [[Sequence|sequence]]:<ref name="C"/>
* {{M|1=\pi=[\pi_1,\pi_2,\cdots,\pi_n]}} which is chosen such that {{M|\pi_1\ge\pi_2\ge\cdots\ge\pi_m > 0}}
As the order is insignificant this ordering makes partitions easy to compare by making identical partitions easy to see.

When the same number occurs multiple times the notation of raising to the power of the number of consecutive terms is used to save writing.<ref name="C"/>
* For example: in the {{M|3}} part partition of {{M|6}}: {{M|[4,1,1]}} it is standard to write {{M|[4,1^2]}} instead
===Signifying {{M|\pi}} is a partition of {{M|n}}===
To signify that {{M|\pi}} is a partition of {{M|n}} we may write:
* {{M|\pi\vdash n}}<ref name="C"/>

==Equality of partitions==
Two partitions, {{M|\pi}} and {{M|\pi'}} are considered the same or ''equal''<ref name="C"/> if they:
# Have the same number of parts and sum to the same number (are both {{M|m}}-part partitions of {{M|n}}) and
# {{M|1=\forall i\in\{1,2,\cdots,m\}[\pi_i=\pi'_i]}}
and we write {{M|1=\pi=\pi'}}

==See also==
* [[Composition (combinatorics)]]
* [[Monomial symmetric function]] - the motivation for these partitions

==References==
<references/>

{{Definition|Combinatorics}}