Editing Permutation (section)

=== Canonical cycle notation ===
In some combinatorial contexts it is useful to fix a certain order for the elements in the cycles and of the (disjoint) cycles themselves. [[Miklós Bóna]] calls the following ordering choices the ''canonical cycle notation:''
* in each cycle the ''largest'' element is listed first
* the cycles are sorted in ''increasing'' order of their first element, not omitting 1-cycles

For example, <math display=block>(513)(6)(827)(94)</math> is a permutation of <math>S = \{1, 2, \ldots , 9\}</math> in canonical cycle notation.<ref>{{harvnb|Bona|2012|loc=p.87}} [The book has a typo/error here, as it gives (45) instead of (54).]</ref>

[[Richard P. Stanley|Richard Stanley]] calls this the "standard representation" of a permutation,<ref name="Stanley2012">{{cite book |last=Stanley |first=Richard P. |title=Enumerative Combinatorics: Volume I, Second Edition |publisher=Cambridge University Press |year=2012 |isbn=978-1-107-01542-5 |page=30, Prop 1.3.1}}</ref> and Martin Aigner uses "standard form".<ref name="Aigner2007">{{cite book|first=Martin|last= Aigner|title=A Course in Enumeration|url=https://archive.org/details/courseenumeratio00aign_007|url-access=limited|year=2007|publisher=Springer GTM 238|isbn=978-3-540-39035-0|pages=[https://archive.org/details/courseenumeratio00aign_007/page/n32 24]–25}}</ref> [[Sergey Kitaev]] also uses the "standard form" terminology, but reverses both choices; that is, each cycle lists its minimal element first, and the cycles are sorted in decreasing order of their minimal elements.<ref name="Kitaev2011">{{cite book|first=Sergey |last=Kitaev|title=Patterns in Permutations and Words|year=2011|publisher=Springer Science & Business Media|isbn=978-3-642-17333-2|page=119}}</ref>