Integer partition: Difference between revisions

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{{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]] and theoretical physics fields, 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}}

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===Conjugate and self-conjugate partitions===

{{anchor|Conjugate partitions}}

If we flip the diagram of the partition 6 + 4 + 3 + 1 along its [[main diagonal]], we obtain another partition of 14:

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One can then obtain a [[bijection]] between the set of partitions with distinct odd parts and the set of self-conjugate partitions, as illustrated by the following example:

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