Partition of a set

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. More formally, these "cells" are both collectively exhaustive and mutually exclusive with respect to the set being partitioned.

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) The union of the elements of P is equal to X. (We say the elements of P cover X.)
 * 2) 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 or parts of the partition.

Examples

 * Every singleton set {x} has exactly one partition, namely { {x} }.
 * For any nonempty set X, P = {X} is a partition of X.
 * For any non-empty proper subset A of a set U, this A together with its 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 (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 if 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 Bn, 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 exponential generating function for Bell numbers is


 * $$\sum_{n=0}^\infty\frac{B_n}{n!}z^n=e^{e^z-1}.$$

Bell numbers satisfy the recursion $$B_{n+1}=\sum_{k=0}^n {n\choose k}B_k$$.

The Stirling number of the second kind S(n, k) 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


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