Editing Permutation (section)

===Circular permutations===
{{see also|Cyclic order#Finite cycles}}
Permutations, when considered as arrangements, are sometimes referred to as ''linearly ordered'' arrangements. If, however, the objects are arranged in a circular manner this distinguished ordering is weakened: there is no "first element" in the arrangement, as any element can be considered as the start. An arrangement of distinct objects in a circular manner is called a '''circular permutation'''.<ref>{{harvnb|Brualdi|2010|p=39}}</ref>{{efn|In older texts ''circular permutation'' was sometimes used as a synonym for [[cyclic permutation]], but this is no longer done. See {{harvtxt|Carmichael|1956|p=7}}}} These can be formally defined as [[equivalence classes]] of ordinary permutations of these objects, for the [[equivalence relation]] generated by moving the final element of the linear arrangement to its front.

Two circular permutations are equivalent if one can be rotated into the other. The following four circular permutations on four letters are considered to be the same.

<pre>
     1           4           2           3
   4   3       2   1       3   4       1   2
     2           3           1           4
</pre>

The circular arrangements are to be read counter-clockwise, so the following two are not equivalent since no rotation can bring one to the other.
<pre>
     1          1
   4   3      3   4
     2          2</pre>

There are (''n''&thinsp;–&thinsp;1)! circular permutations of a set with ''n'' elements.