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Definitions: Partition |
PartitionNoun1. A vertical structure that divides or separates (as a wall divides one room from another). 2. The act of dividing or partitioning; separation by the creation of a boundary that divides or keeps apart. 3. (computer science) the part of a hard disk that is dedicated to a particular operating system or application and accessed as a single unit. Verb1. Divide into parts, pieces, or sections; "The Arab peninsula was partitioned by the British". 2. Separate or apportion into sections; "partition a room off". Source: WordNet 1.7.1 Copyright © 2001 by Princeton University. All rights reserved. |
Date "partition" was first used in popular English literature: sometime before 1010. (references) |
| Domain | Definitions |
Computing | Partition 1. |
Building & Civil Engineering | The term partition is applied to walls, either loadbearing or non loadbearing, dividing the space within a building into rooms. Source: European Union. (references) |
Census | A portion of the TIGER(r) database separated to effectively manage the size of that database in order to support operations such as updating, processing, and mapping of a specific part of the database. A partition usually consists of an entire county or statistically equivalent entity, but a county that has many records in the database may be divided into multiple partitions to allow the computer to process, and enable staff to work with, smaller files. Also referred to as a county partiti. (references) |
General | Movable interior wall. Source: European Union. (references) |
Law | The forced division of land among parties who were formerly co-owners. Source: European Union. (references) |
Math | (1) A division of a set into nonempty disjoint sets which completely cover the set. (2) To rearrange the elements of an array into two (or more) groups, typically, such that elements in the first group are less than a value and elements in the second group are greater. (references) |
Mechanical Engineering | Any part delimiting a hollow space in a machine. Source: European Union. (references) |
Transportation | Wall in a structure which serves to strenghten, divide or help give shape to the structure. Source: European Union. (references) |
Source: compiled by the editor from various references; see credits. | |
(From Wikipedia, the free Encyclopedia)
The partitions of 4 are listed below:
The partition 6 + 4 + 3 + 1 of the positive number 14 can be represented
by the following graph:
Claim: The number of self-conjugate partitions is the same as the
number of partitions with distinct odd parts.
Proof (sketch): The crucial observation is that every odd part can
be "folded" in the middle to form a self conjugate graph:
The number of partitions of a positive integer n is given by the partition function p(n). The number of partitions of n into
exactly k parts is denoted by pk(n).
Ferrers graph techniques also allow us to prove results like the following:
Examples
The partitions of 8 are listed below:
Among the 22 partitions for the number 8, 6 of them contain
only odd parts:
Curiously, if we count the partitions of 8
with distinct parts, we also obtain the number 6:
Is this only coincidence, or is it true that, for all positive number,
the number of partitions with odd parts always equals the number
of partitions with distinct parts? This and other results
can be obtained by the aid of a visual tool, Ferrers' graph.Ferrers' Graph
o o o o
o o o
o o o
o o
o
o
6+4+3+1
The 14 circles are lined up in 4 columns, each having the size of a part
of the partition. The graphs for the 5 partitions of the number 4 are
listed below:
o o o o o o o o o o o o
o o o o o
o o
o
4 3+1 2+2 2+1+1 1+1+1+1
If we now flip the graph of the partition 6 + 4 + 3 + 1 along the
NW-SE axis, we obtain another partition of 14:
o o o o o o o o o o
o o o o o o o
o o o --> o o o
o o o
o
o
6+4+3+1 4+3+3+2+1+1
By turning the rows into columns, we obtain the partition 4 + 3 + 3 + 2 +
1 + 1 of the number 14. Such partitions are said to be
conjugate of one another. In the case of the number 4, partitions
4 and 1 + 1 + 1 + 1 are conjugate pairs, and partitions 3 + 1 and 2 + 1 + 1
are conjugate of each other. Of particular interest is the partition
2 + 2, which has itself as conjugate. Such a paritition is said to be
self-conjugate.o
o
o --> o o o
o o
o o
One can then obtain a bijection between the set of partitions with distinct
odd parts and the set of self-conjugate partitions, as illustrated by
the following example:
o * x o o o o o
o * x o * * * *
o * x <--> o * x x
o * o * x
o * o *
o *
o *
o
o
9+7+3 5+5+4+3+2
distinct odd self-conjugateSimilar techniques can be employed to establish, for example, the following
equalities:
Number of Partitions
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