The generalized faro shuffle

The standard faro shuffle, an idealized riffle shuffle, divides the deck into two equal portions, and perfectly interlaces them. The simple cut takes one card from the top to the bottom of the deck. It is known that for decks of even size, the faro shuffle and simple cut generate all possible permutations, while if the deck is of odu size, only a small fraction are available. This paper considers a generalized faro shuffle wherein the deck is divided into n rather than 2, portions and these portions are “interlaced” together. It is shown that the generalized faro shuffle and the simple cut generate either the symmetric group of the deck, the alternating group of the deck, or in one special case, only a small fraction of the possible permutations. Whether the symmetric group or alternating group is generated depends on the parity of the simple cut and the generalized faro shuffle as group operations.