Editing Integer partition (section)

==Rank and Durfee square==
{{main|Durfee square}}
The ''rank'' of a partition is the largest number ''k'' such that the partition contains at least ''k'' parts of size at least ''k''.  For example, the partition&nbsp;4&nbsp;+&nbsp;3&nbsp;+&nbsp;3&nbsp;+&nbsp;2&nbsp;+&nbsp;1&nbsp;+&nbsp;1 has rank 3 because it contains 3 parts that are ≥&nbsp;3, but does not contain 4 parts that are&nbsp;≥&nbsp;4.  In the Ferrers diagram or Young diagram of a partition of rank ''r'', the ''r'' × ''r'' square of entries in the upper-left is known as the [[Durfee square]]:

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

The Durfee square has applications within combinatorics in the proofs of various partition identities.<ref>see, e.g., {{harvnb|Stanley|1999|p=58}}</ref>  It also has some practical significance in the form of the [[h-index]].

A different statistic is also sometimes called the [[rank of a partition]] (or Dyson rank), namely, the difference <math>\lambda_k - k</math> for a partition of ''k'' parts with largest part <math>\lambda_k</math>.  This statistic (which is unrelated to the one described above) appears in the study of [[Ramanujan congruences]].