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# Partition (mathematics)

In mathematics, **partition** refers to two related concepts, in set theory and number theory.

## Partition (set theory)

A *partition* of a set *X* is a collection of non-empty subsets ("parts") of *X* such that every element of *X* is in exactly one of the subsets in .

Hence a three-element set {*a*,*b*,*c*} has 5 partitions:

- {
*a*,*b*,*c*} - {
*a*,*b*}, {*c*} - {
*a*,*c*}, {*b*} - {
*b*,*c*}, {*a*} - {
*a*}, {*b*}, {*c*}

Partitions and equivalence relations give the same data: the equivalence classes of an equivalence relation on a set *X* form a partition of the set *X*, and a partition gives rise to an equivalence relation where two elements are equivalent if they are in the same part from .

The number of partitions of a finite set of size *n* into *k* parts is given by a Stirling number of the second kind;

### Bell numbers

The total number of partitions of a set of size *n* is given by the *n*-th **Bell number**, denoted *B*_{n}. These may be obtained by the recurrence relation

They have an exponential generating function

Asymptotically,

where *W* denotes the Lambert W function.

## Partition (number theory)

A *partition* of an integer *n* is an expression of *n* as a sum of positive integers ("parts"), with the order of the terms in the sum being disregarded.

Hence the number 3 has 3 partitions:

- 3
- 2+1
- 1+1+1

The number of partitions of *n* is given by the partition function *p*(*n*).

## References

- Chen Chuan-Chong; Koh Khee-Meng (1992).
*Principles and Techniques of Combinatorics*. World Scientific. ISBN 981-02-1139-2.