Integer partition: Difference between revisions

Revision as of 22:30, 27 March 2025

Revision as of 05:30, 7 April 2025

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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>.