Faro Shuffle - Group Theory Aspects

Group Theory Aspects

In mathematics, a perfect shuffle can be considered to be an element of the symmetric group.

More generally, in, the perfect shuffle is the permutation that splits the set into 2 piles and interleaves them:

$\begin{pmatrix} 1 & 2 & 3 & 4 & \cdots \\ 1 & n+1 & 2 & n+2 & \cdots \end{pmatrix}$

Formally, it sends

$k \mapsto \begin{cases} 2k-1 & k\leq n\\ 2(k-n) & k> n \end{cases}$

Analogously, the -perfect shuffle permutation is the element of that splits the set into k piles and interleaves them.

The -perfect shuffle, denote it, is the composition of the -perfect shuffle with an -cycle, so the sign of is:

The sign is thus 4-periodic:

$\mbox{sgn}(\rho_n) = (-1)^{\lfloor n/2 \rfloor} = \begin{cases} +1 & n \equiv 0,1 \pmod{4}\\ -1 & n \equiv 2,3 \pmod{4} \end{cases}$

The first few perfect shuffles are: and are trivial, and is the transposition .

Read more about this topic: Faro Shuffle

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