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

Famous quotes containing the words group, theory and/or aspects:

    “The boys think they can all be athletes, and the girls think they can all be singers. That’s the way to fame and success. ...as a group blacks must give up their illusions.”
    —Kristin Hunter (b. 1931)

    “In the theory of gender I began from zero. There is no masculine power or privilege I did not covet. But slowly, step by step, decade by decade, I was forced to acknowledge that even a woman of abnormal will cannot escape her hormonal identity.”
    —Camille Paglia (b. 1947)

    “Grammar is a tricky, inconsistent thing. Being the backbone of speech and writing, it should, we think, be eminently logical, make perfect sense, like the human skeleton. But, of course, the skeleton is arbitrary, too. Why twelve pairs of ribs rather than eleven or thirteen? Why thirty-two teeth? It has something to do with evolution and functionalism—but only sometimes, not always. So there are aspects of grammar that make good, logical sense, and others that do not.”
    —John Simon (b. 1925)