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.

Contents

Definition

Examples

Partitions and equivalence relations

Partial ordering of the lattice of partitions

Noncrossing partitions

The number of partitions

See also

Notes

References

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 include 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' or 'parts' of the partition.^[1]

Examples

★ Every singleton set {''x''} has exactly one partition, namely { {''x''} }.

★ For any nonempty set ''X'', ''P'' = {''X''} is a partition of ''X''.

★ The empty set has exactly one partition, namely one with no blocks.

★ For any non-empty proper subset ''A'' of a set ''U'', this ''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 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'' if there exists a member of ''P'' which contains both ''x'' and ''y''. The notions of "equivalence relation" and "partition" are thus essentially equivalent.^[2]

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 ''B''₀ = 1,
''B''₁ = 1, ''B''₂ = 2, ''B''₃ = 5, ''B''₄ = 15, ''B''₅ = 52, ''B''₆ = 203.
The exponential generating function for Bell numbers is
:
Bell numbers satisfy the recursion $B\_\{n+1\}=sum\_\{k=0\}^n\; \{nchoose\; k\}B\_k$.
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 ''n''th Catalan number, given by
:$C\_n=\{1\; over\; n+1\}\{2n\; choose\; n\}.$

See also

★ Data clustering

★ Equivalence relation

★ Exponential formula

★ List of partition topics

★ Partial equivalence relation

Notes

1. Brualdi, ''pp''. 44-45
2. Schechter, ''p''. 54

References

★ Introductory Combinatorics, , Richard A., Brualdi, Pearson Prentice Hall, 2004,

★ Handbook of Analysis and Its Foundations, , Eric, Schechter, Academic Press, 1997,