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 B_n. 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).

References

• Chen Chuan-Chong; Koh Khee-Meng (1992). Principles and Techniques of Combinatorics. World Scientific. ISBN 981-02-1139-2.