[1] A Gilbreath shuffle consists of the following two steps:[1] It differs from the more commonly used procedure of cutting a deck into two piles and then riffling the piles, in that the first step of dealing off cards reverses the order of the cards in the new pile, whereas cutting the deck would preserve this order.
Although seemingly highly random, Gilbreath shuffles preserve many properties of the initial deck.
This phenomenon is known as Gilbreath's principle and is the basis for several card tricks.
[1] Mathematically, Gilbreath shuffles can be described by Gilbreath permutations, permutations of the numbers from 1 to n that can be obtained by a Gilbreath shuffle with a deck of cards labeled with these numbers in order.
Gilbreath permutations can be characterized by the property that every prefix contains a consecutive set of numbers.
[2] A Gilbreath shuffle may be uniquely determined by specifying which of the positions in the resulting shuffled deck are occupied by cards that were dealt off into the second pile, and which positions are occupied by cards that were not dealt off.
possible ways of performing a Gilbreath shuffle on a deck of
[1][3] The cyclic Gilbreath permutations of order
are in one-to-one correspondence with the real numbers
describes the numerical sorted order of the iterates for
[1] The number of cyclic Gilbreath permutations (and therefore also the number of real periodic points of the Mandelbrot set), for
, is given by the integer sequence A theorem called "the ultimate Gilbreath principle" states that, for a permutation