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:

    “Now, honestly: if a large group of ... demonstrators blocked the entrances to St. Patrick’s Cathedral every Sunday for years, making it impossible for worshipers to get inside the church without someone escorting them through screaming crowds, wouldn’t some judge rule that those protesters could keep protesting, but behind police lines and out of the doorways?”
    —Anna Quindlen (b. 1953)

    “No theory is good unless it permits, not rest, but the greatest work. No theory is good except on condition that one use it to go on beyond.”
    —André Gide (1869–1951)

    “That anger can be expressed through words and non-destructive activities; that promises are intended to be kept; that cleanliness and good eating habits are aspects of self-esteem; that compassion is an attribute to be prized—all these lessons are ones children can learn far more readily through the living example of their parents than they ever can through formal instruction.”
    —Fred Rogers (20th century)