Integer partition: Difference between revisions

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* 1 + 1 + 1 + 1 + 1

Some authors treat a partition as a decreasingnon-increasing sequence of summands, rather than an expression with plus signs. For example, the partition 2 + 2 + 1 might instead be written as the [[tuple]] {{math|(2, 2, 1)}} or in the even more compact form {{math|(2², 1)}} where the superscript indicates the number of repetitions of a part.

This multiplicity notation for a partition can be written alternatively as <math>1^{m_1}2^{m_2}3^{m_3}\cdots</math>, where {{math|''m''₁}} is the number of 1's, {{math|''m''₂}} is the number of 2's, etc. (Components with {{math|''m''_''i'' {{=}} 0}} may be omitted.) For example, in this notation, the partitions of 5 are written <math>5^1, 1^1 4^1, 2^1 3^1, 1^2 3^1, 1^1 2^2, 1^3 2^1</math>, and <math>1^5</math>.

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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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* {{cite web|last1=Grime|first1=James|title=Partitions - Numberphile|url=https://www.youtube.com/watch?v=NjCIq58rZ8I| archive-url=https://ghostarchive.org/varchive/youtube/20211211/NjCIq58rZ8I| archive-date=2021-12-11 | url-status=live|publisher=[[Brady Haran]]|access-date=5 May 2016|format=video|date=April 28, 2016}}{{cbignore}}

{{DEFAULTSORT:Partition (Number Theory)}}

[[Category:Integer partitions| ]]