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Concept
Partition (number theory)
Summary
In number theory and combinatorics, a partition of a non-negative integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. (If order matters, the sum becomes a composition.) For example, 4 can be partitioned in five distinct ways:
4
3 + 1
2 + 2
2 + 1 + 1
1 + 1 + 1 + 1
The only partition of zero is the empty sum, having no parts.
The order-dependent composition 1 + 3 is the same partition as 3 + 1, and the two distinct compositions 1 + 2 + 1 and 1 + 1 + 2 represent the same partition as 2 + 1 + 1.
An individual summand in a partition is called a part. The number of partitions of n is given by the partition function p(n). So p(4) = 5. The notation λ ⊢ n means that λ is a partition of n.
Partitions can be graphically visualized with Young diagrams or Ferrers diagrams. They occur in a number of branches of mathematics and physics, including the study of symmetric polynomials and of the symmetric group and in group representation theory in general.
The seven partitions of 5 are
5
4 + 1
3 + 2
3 + 1 + 1
2 + 2 + 1
2 + 1 + 1 + 1
1 + 1 + 1 + 1 + 1
Some authors treat a partition as a decreasing sequence of summands, rather than an expression with plus signs. For example, the partition 2 + 2 + 1 might instead be written as the tuple (2, 2, 1) or in the even more compact form (22, 1) where the superscript indicates the number of repetitions of a part.
This multiplicity notation for a partition can be written alternatively as , where m1 is the number of 1's, m2 is the number of 2's, etc. (Components with mi = 0 may be omitted.) For example, in this notation, the partitions of 5 are written , and .
There are two common diagrammatic methods to represent partitions: as Ferrers diagrams, named after Norman Macleod Ferrers, and as Young diagrams, named after Alfred Young. Both have several possible conventions; here, we use English notation, with diagrams aligned in the upper-left corner.
Official source
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