Editing Symmetric group (section)

=== Special elements ===
Certain elements of the symmetric group of {1, 2, ..., ''n''} are of particular interest (these can be generalized to the symmetric group of any finite totally ordered set, but not to that of an unordered set).

The '''{{visible anchor|order reversing permutation}}''' is the one given by:
:<math>\begin{pmatrix} 1 & 2 & \cdots & n\\
n & n-1 & \cdots & 1\end{pmatrix}.</math>
This is the unique maximal element with respect to the [[Bruhat order]] and the
[[longest element of a Coxeter group|longest element]] in the symmetric group with respect to generating set consisting of the adjacent transpositions {{nowrap|(''i'' ''i''+1)}}, {{nowrap|1 ≤ ''i'' ≤ ''n'' − 1}}.

This is an involution, and consists of <math>\lfloor n/2 \rfloor</math> (non-adjacent) transpositions
:<math>(1\,n)(2\,n-1)\cdots,\text{ or }\sum_{k=1}^{n-1} k = \frac{n(n-1)}{2}\text{ adjacent transpositions: }</math>
:: <math>(n\,n-1)(n-1\,n-2)\cdots(2\,1)(n-1\,n-2)(n-2\,n-3)\cdots,</math>

so it thus has sign:

:<math>\mathrm{sgn}(\rho_n) = (-1)^{\lfloor n/2 \rfloor} =(-1)^{n(n-1)/2} = \begin{cases}
+1 & n \equiv 0,1 \pmod{4}\\
-1 & n \equiv 2,3 \pmod{4}
\end{cases}</math>
which is 4-periodic in ''n''.

In S<sub>2''n''</sub>, the ''[[Faro shuffle|perfect shuffle]]'' is the permutation that splits the set into 2 piles and interleaves them. Its sign is also <math>(-1)^{\lfloor n/2 \rfloor}.</math>

Note that the reverse on ''n'' elements and perfect shuffle on 2''n'' elements have the same sign; these are important to the classification of [[Clifford algebra]]s, which are 8-periodic.