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:

    “Just as a person who is always asserting that he is too good-natured is the very one from whom to expect, on some occasion, the coldest and most unconcerned cruelty, so when any group sees itself as the bearer of civilization this very belief will betray it into behaving barbarously at the first opportunity.”
    —Simone Weil (1910–1943)

    “Freud was a hero. He descended to the “Underworld” and met there stark terrors. He carried with him his theory as a Medusa’s head which turned these terrors to stone.”
    —R.D. (Ronald David)

    “... of all the aspects of social misery nothing is so heartbreaking as unemployment ...”
    —Jane Addams (1860–1935)