Partition of a set

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A partition of U into 6 blocks:
a Venn diagram representation.

In mathematics, a partition of a set X is a division of X into non-overlapping "parts" or "blocks" or "cells" that cover all of X.

Contents

1 Definition

2 Examples

3 Partitions and equivalence relations

4 Partial ordering of the lattice of partitions

5 Noncrossing partitions

6 The number of partitions

7 See also

Definition

A partition of a set X is a set of nonempty subsets of X such that every element x in X is in exactly one of these subsets.

Equivalently, a set P of subsets of X, is a partition of X if

No element of P is empty. (NB - some definitions do not require this)
The union of the elements of P is equal to X. (We say the elements of P cover X.)
The intersection of any two elements of P is empty. (We say the elements of P are pairwise disjoint.)

The elements of P are sometimes called the blocks of the partition.

Examples

Every singleton set {x} has exactly one partition, namely { {x} }.
For any set X, P = {X} is a partition of X.
The empty set has exactly one partition, namely one with no blocks.
Forgetting momentarily about certain exotic cases, the set of all humans can be partitioned into two blocks: the males and the females.
For any non-empty proper subset A of a set U, then A together with its complement is a partition of U.
If we do not use axiom 1, then the above example generalizes so that any subset (empty or not) together with its complement is a partition.
The set { 1, 2, 3 } has these five partitions.
- { {1}, {2}, {3} }, sometimes denoted by 1/2/3.
- { {1, 2}, {3} }, sometimes denoted by 12/3.
- { {1, 3}, {2} }, sometimes denoted by 13/2.
- { {1}, {2, 3} }, sometimes denoted by 1/23.
- { {1, 2, 3} }, sometimes denoted by 123.
Note that
- { {}, {1,3}, {2} } is not a partition if we are using axiom 1 (because it contains the empty set); otherwise it is a a partition of {1, 2, 3}.
- { {1,2}, {2, 3} } is not a partition (of any set) because the element 2 is contained in more than one distinct subset.
- { {1}, {2} } is not a partition of {1, 2, 3} because none of its blocks contains 3; however, it is a partition of {1, 2}.

Partitions and equivalence relations

If an equivalence relation is given on the set X, then the set of all equivalence classes forms a partition of X. Conversely, if a partition P is given on X, we can define an equivalence relation on X by writing x ~ y iff there exists a member of P which contains both x and y. The notions of "equivalence relation" and "partition" are thus essentially equivalent.

Partial ordering of the lattice of partitions

Given two partitions π and ρ of a given set X, we say that π is finer than ρ, or, equivalently, that ρ is coarser than π, if π splits the set X into smaller blocks than ρ does, i.e. if every element of π is a subset of some element of ρ. In that case, one writes π ≤ ρ.

The relation of "being-finer-than" is a partial order on the set of all partitions of the set X, and indeed even a complete lattice. In case n = 4, the partial order of the set of all 15 partitions is depicted in this Hasse diagram:

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Noncrossing partitions

The lattice of noncrossing partitions of a finite set has recently taken on importance because of its role in free probability theory. These form a subset of the lattice of all partitions, but not a sublattice, since the join operations of the two lattices do not agree.

The number of partitions

The Bell number B_n, named in honor of Eric Temple Bell, is the number of different partitions of a set with n elements. The first several Bell numbers are B₀ = 1, B₁ = 1, B₂ = 2, B₃ = 5, B₄ = 15, B₅ = 52, B₆ = 203.

The Stirling number S(n, k) of the second kind is the number of partitions of a set of size n into k blocks.

The number of partitions of a set of size n corresponding to the integer partition

$n=\underbrace{1+\cdots+1}_{m_1\ \mbox{terms}} +\underbrace{2+\cdots+2}_{m_2\ \mbox{terms}} +\underbrace{3+\cdots+3}_{m_3\ \mbox{terms}}+\cdots$

of n, is the Faà di Bruno coefficient

${n! \over m_1!m_2!m_3!\cdots 1!^{m_1}2!^{m_2}3!^{m_3}\cdots}.$

The number of noncrossing partitions of a set of size n is the nth Catalan number, given by

$C_n={1 \over n+1}{2n \choose n}.$