Editing Integer partition (section)

===Conjugate and self-conjugate partitions===
{{anchor|Conjugate partitions}}
If we flip the diagram of the partition 6 + 4 + 3 + 1 along its [[main diagonal]], we obtain another partition of 14:

{|
|- style="vertical-align:top; text-align:left;"
| [[File:RedDot.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:RedDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]]<br />[[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:RedDot.svg|16px|*]]<br />[[File:GrayDot.svg|16px|*]]
| style="vertical-align:middle;"| ↔
| [[File:RedDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]]<br />[[File:GrayDot.svg|16px|*]][[File:RedDot.svg|16px|*]][[File:GrayDot.svg|16px|*]]<br />[[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:RedDot.svg|16px|*]]<br />[[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]]<br />[[File:GrayDot.svg|16px|*]]<br />[[File:GrayDot.svg|16px|*]]
|- style="vertical-align:top; text-align:center;"
| 6 + 4 + 3 + 1
| =
| 4 + 3 + 3 + 2 + 1 + 1
|}

By turning the rows into columns, we obtain the partition 4&nbsp;+&nbsp;3&nbsp;+&nbsp;3&nbsp;+&nbsp;2&nbsp;+&nbsp;1&nbsp;+&nbsp;1 of the number 14. Such partitions are said to be ''conjugate'' of one another.{{sfn|Hardy|Wright|2008|p=362}} In the case of the number 4, partitions 4 and 1&nbsp;+&nbsp;1&nbsp;+&nbsp;1&nbsp;+&nbsp;1 are conjugate pairs, and partitions 3&nbsp;+&nbsp;1 and 2&nbsp;+&nbsp;1&nbsp;+&nbsp;1 are conjugate of each other. Of particular interest are partitions, such as 2&nbsp;+&nbsp;2, which have themselves as conjugate. Such partitions are said to be ''self-conjugate''.{{sfn|Hardy|Wright|2008|p=368}}

'''Claim''': The number of self-conjugate partitions is the same as the number of partitions with distinct odd parts.

'''Proof (outline)''': The crucial observation is that every odd part can be "''folded''" in the middle to form a self-conjugate diagram:

{|
|-
| style="vertical-align:top;"| [[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:RedDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]]
| style="vertical-align:top;"| &nbsp; ↔ &nbsp;
| style="vertical-align:top;"| [[File:RedDot.svg|16px|*]][[File:GrayDot.svg|16px|*]][[File:GrayDot.svg|16px|*]]<br />[[File:GrayDot.svg|16px|*]]<br />[[File:GrayDot.svg|16px|*]]
|}

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:

{|
|-
| valign="top" | 
[[File:GrayDot.svg|16px|o]][[File:GrayDot.svg|16px|o]][[File:GrayDot.svg|16px|o]][[File:GrayDot.svg|16px|o]][[File:GrayDot.svg|16px|o]][[File:GrayDot.svg|16px|o]][[File:GrayDot.svg|16px|o]][[File:GrayDot.svg|16px|o]][[File:GrayDot.svg|16px|o]]
<br />
[[File:RedDot.svg|16px|*]][[File:RedDot.svg|16px|*]][[File:RedDot.svg|16px|*]][[File:RedDot.svg|16px|*]][[File:RedDot.svg|16px|*]][[File:RedDot.svg|16px|*]][[File:RedDot.svg|16px|*]]
<br />
[[File:BlackDot.svg|16px|x]][[File:BlackDot.svg|16px|x]][[File:BlackDot.svg|16px|x]]
| style="vertical-align:middle;"| ↔
| valign="top" | [[File:GrayDot.svg|16px|o]][[File:GrayDot.svg|16px|o]][[File:GrayDot.svg|16px|o]][[File:GrayDot.svg|16px|o]][[File:GrayDot.svg|16px|o]]<br />[[File:GrayDot.svg|16px|o]][[File:RedDot.svg|16px|*]][[File:RedDot.svg|16px|*]][[File:RedDot.svg|16px|*]][[File:RedDot.svg|16px|*]]<br />[[File:GrayDot.svg|16px|o]][[File:RedDot.svg|16px|*]][[File:BlackDot.svg|16px|x]][[File:BlackDot.svg|16px|x]]<br />[[File:GrayDot.svg|16px|o]][[File:RedDot.svg|16px|*]][[File:BlackDot.svg|16px|x]]<br />[[File:GrayDot.svg|16px|o]][[File:RedDot.svg|16px|*]]
|- style="vertical-align:top; text-align:center;"
| 9 + 7 + 3
| =
| 5 + 5 + 4 + 3 + 2
|- style="vertical-align:top; text-align:center;"
| Dist. odd
|
| self-conjugate
|}