integer partition

An (unordered) partition of a natural number $n$ is a way of writing $n$ as a sum of natural numbers. For example, the following are the all partitions of $5$ :

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 $\{8,5,5,5,2,1,1\}$ for $27=8+5+5+5+2+1+1$ . Another common notation is to write multiplicities as superscripts, such as $1^22^15^38$ for $27=8+5+5+5+2+1+1$ . Note that this way of writing partitions typically uses smallest to largest.

Partitions are often drawn as Young diagrams, which are rectangular arrays of boxes in which the $k$ 'th row has a number of boxes equal to the $k$ 'th part of the partition. Sometimes dots are used instead of boxes, and then the obtained picture is called a Ferrers diagram. For instance, the partition $10=5+2+2+1$ is drawn

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 $10=4+3+1+1+1$ .

References

1

Richard P. Stanley.
Enumerative Combinatorics, volume I.
Wadsworth & Brooks, 1986.
Zbl 0608.05001.