Partition of a set

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

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

1. No element of P is empty.
2. The union of the elements of P is equal to X. (We say the elements of P cover X.)
3. 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} }.
• The empty set has exactly one partition, namely itself. (Axioms 1 and 3 are vacuously satisfied.)
• 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 subset A of a set U, then A together with it's complement is a partition of U.
• 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 (of any set) because it contains the empty set.
  • { {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:

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

<math>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<math>

of n, is the noncrossing partitions (which see) of a set of size n is the nth Catalan number, given by

<math>C_n={1 \over n+1}{2n \choose n}.<math>