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Partition (mathematics)

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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 Bn. 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