partions

Partions, usually written partitions, is a term used in several fields to describe a division of a whole into separate parts. It can refer to mathematical groupings, the organization of storage devices, or other contexts in which an object is split into components. The spelling partions is a common error.

In set theory, a partition of a set S is a collection of nonempty, pairwise disjoint subsets whose union is S. For S = {a, b, c}, the five partitions are { {a}, {b}, {c} }, { {a,b}, {c} }, { {a,c}, {b} }, { {b,c}, {a} }, and { {a,b,c} }. The Bell numbers count the total number of partitions of an n-element set; for n=3, B3=5 and for n=4, B4=15.

An integer partition of n is a way of writing n as a sum of positive integers in nonincreasing order. Examples for n=5 are: 5; 4+1; 3+2; 3+1+1; 2+2+1; 2+1+1+1; 1+1+1+1+1. The partition function p(n) counts such partitions; p(5)=7. Partitions differ from compositions, where order of terms matters.

Disk partitions are regions of a physical drive designated to hold a file system or data. A drive may be partitioned into primary and, in older schemes, extended and logical partitions, while modern GPT disks use a set of independent partitions. Partition tables (MBR and GPT) describe the layout. Disk partitioning is performed by tools such as fdisk, gdisk, or parted and is used for booting, multi-boot configurations, or data organization.

Partitions as a concept appear across mathematics, computer science, and information technology, illustrating a general idea of dividing a whole into meaningful parts.