Partition (mathematics)

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 $$\mathcal{P}$$ 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:


 * {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 $$\mathcal{P}$$ 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;

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


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

Asymptotically,


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

where W denotes the Lambert W function.

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).