Editing Integer partition (section)

==Diagrammatic representations of partitions==
There are two common diagrammatic methods to represent partitions: as Ferrers diagrams, named after [[Norman Macleod Ferrers]], and as Young diagrams, named after [[Alfred Young (mathematician)|Alfred Young]].  Both have several possible conventions; here, we use ''English notation'', with diagrams aligned in the upper-left corner.

===Ferrers diagram <!-- [[Ferrers diagram]] and [[Ferrers diagrams]] currently redirect to this section. --> ===
The partition 6&nbsp;+&nbsp;4&nbsp;+&nbsp;3&nbsp;+&nbsp;1 of the number 14 can be represented by the following diagram:

[[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]]<br/>[[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]]<br />[[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]]<br />[[File:GrayDot.svg|16px|*]]

The 14 circles are lined up in 4 rows, each having the size of a part of the partition.
The diagrams for the 5 partitions of the number 4 are shown below:

{|
|- style="vertical-align:top; text-align:left;"
| [[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]] ||
| [[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]]<br />[[File:GrayDot.svg|16px|*]] ||
| [[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]]<br />[[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]] ||
| [[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]]<br />[[File:GrayDot.svg|16px|*]]<br />[[File:GrayDot.svg|16px|*]] ||
| [[File:GrayDot.svg|16px|*]]<br />[[File:GrayDot.svg|16px|*]]<br />[[File:GrayDot.svg|16px|*]]<br />[[File:GrayDot.svg|16px|*]]
|- style="vertical-align:top; text-align:center;"
| 4 || =
| 3 + 1 || =
| 2 + 2 || =
| 2 + 1 + 1 || =
| 1 + 1 + 1 + 1
|}

===Young diagram===
{{Main|Young diagram}}

An alternative visual representation of an integer partition is its ''Young diagram'' (often also called a Ferrers diagram).  Rather than representing a partition with dots, as in the Ferrers diagram, the Young diagram uses boxes or squares.  Thus, the Young diagram for the partition 5 + 4 + 1 is
:[[Image:Young diagram for 541 partition.svg|100px]]
while the Ferrers diagram for the same partition is
:{|
|- style="vertical-align:top; text-align:left;"
| [[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]]<br />[[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]]<br />[[File:GrayDot.svg|16px|*]]
|- style="vertical-align:top; text-align:center;"
|}

While this seemingly trivial variation does not appear worthy of separate mention, Young diagrams turn out to be extremely useful in the study of [[symmetric functions]] and [[group representation theory]]: filling the boxes of Young diagrams with numbers (or sometimes more complicated objects) obeying various rules leads to a family of objects called [[Young tableaux]], and these tableaux have combinatorial and representation-theoretic significance.{{sfn|Andrews|1976|p=199}} As a type of shape made by adjacent squares joined together, Young diagrams are a special kind of [[polyomino]].<ref>{{citation
 | last = Josuat-Vergès | first = Matthieu
 | arxiv = 0801.4928
 | doi = 10.1016/j.jcta.2010.03.006
 | issue = 8
 | journal = [[Journal of Combinatorial Theory]]
 | mr = 2677686
 | pages = 1218–1230
 | series = Series A
 | title = Bijections between pattern-avoiding fillings of Young diagrams
 | volume = 117
 | year = 2010| s2cid = 15392503
 }}.</ref>