Partition (mathematics)

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In mathematics, partition refers to two related concepts, in set theory and number theory.


[edit] Partition (set theory)

A partition of a set X is a collection \mathcal{P} of non-empty subsets ("parts") of X such that every element of X is in exactly one of the subsets in \mathcal{P}.

Hence a three-element set {a,b,c} has 5 partitions:

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 \mathcal{P} gives rise to an equivalence relation where two elements are equivalent if they are in the same part from \mathcal{P}.

The number of partitions of a finite set of size n into k parts is given by a Stirling number of the second kind;

[edit] 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

B_n=\sum_{k=0}^{n-1} \binom{m-1}{k} B_k .

They have an exponential generating function

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


 B_n \sim \frac{n!}{\sqrt{2\pi W^2(n)e^{W(n)}}}\frac{e^{e^{W(n)}-1}}{W^n(n)},

where W denotes the Lambert W function.

[edit] 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:

The number of partitions of n is given by the partition function p(n).

[edit] References

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