Partition (mathematics)

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

Partition (set theory)[edit]

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

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

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[edit]

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

${\displaystyle B_n={\displaystyle \sum _{k=0}^{n-1}}{\displaystyle (\genfrac{}{}{0ex}{}{m-1}{k})}{B}_k.}$

They have an exponential generating function

${\displaystyle e^{e^x-1}={\displaystyle \sum _{n=0}^{\mathrm{\infty }}}\frac{B_n}{n!}{x}^n.}$

Asymptotically,

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

where W denotes the Lambert W function.

Partition (number theory)[edit]

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[edit]

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