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integer partition
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(Definition)
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An (unordered) partition of a natural number  is a way of writing  as a sum of natural numbers. For example, the following are the all partitions of  :
Conventionally, parts of a partition are written from the largest to the smallest. Instead of writing the partition as a sum, it is common to use the multiset notation, such as
 for
 . Another common notation is to write multiplicities as superscripts, such as
 for
 .
Partitions are often drawn as Young diagrams which are rectangular arrays of boxes in which  'th row has the number of boxes equal to the  'th part of the partition. Sometimes dots are used instead of boxes, and then the obtained picture is called Ferrers diagram. For instance, the partition
 is drawn like
The dual partition is the partition obtained by reflecting the Young diagram along the main diagonal. For example, the Young diagram of the partition dual to the one above is
which is the diagram of
 .
- 1
- Richard P. Stanley.
Enumerative Combinatorics, volume I.
Wadsworth & Brooks, 1986.
Zbl 0608.05001.
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"integer partition" is owned by bbukh.
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(view preamble)
See Also: partition function
| Other names: |
partition, unordered partition |
| Also defines: |
dual partition, Young diagram, Ferrers diagram |
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Cross-references: diagonal, superscripts, multiplicities, multiset, parts of a partition, sum, natural number
There are 46 references to this entry.
This is version 4 of integer partition, born on 2004-04-10, modified 2006-06-09.
Object id is 5748, canonical name is IntegerPartition.
Accessed 13606 times total.
Classification:
| AMS MSC: | 05A17 (Combinatorics :: Enumerative combinatorics :: Partitions of integers) | | | 11P99 (Number theory :: Additive number theory; partitions :: Miscellaneous) |
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Pending Errata and Addenda
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