faro: Faro shuffles

In permutations: The Symmetric Group: Permutations of a Finite Set

Faro shuffles

Description

A faro shuffle, faro(),is a permutation of a deck of 2n cards. The cards are split into two packs, 1:n and (n+1):2n, and interleaved: cards are taken alternately from top of each pack and placed face down on the table. A faro out-shuffle takes the first card from 1:n and a faro in-shuffle takes the first card from (n+1):(2*n).

A generalized faro shuffle, faro_gen(), splits the pack into m equal parts and applies the same permutation (p1) to each pack, and then permutes the packs with p2, before interleaving. The interleaving itself is simply a matrix transpose; it is possible to omit this step by passing interleave=FALSE.

Usage

faro(n, out = TRUE)
faro_gen(n,m,p1=id,p2=id,interleave=TRUE)

Arguments

n	Number of cards in each pack
m	Number of packs
p1, p2	Permutations for cards and packs respectively, coerced to word form
interleave	Boolean, with default TRUE meaning to actually perform the interleaving and FALSE meaning not to
out	Boolean, with default TRUE meaning to return an out-shuffle and FALSE meaning to return an in-shuffle

Value

Returns a permutation in word form

Author(s)

Robin K. S. Hankin

Examples

faro(4)
faro(4,FALSE)

## Do a perfect riffle shuffle 52 times, return pack to original order:
permorder(faro(26))

## 15 cards, split into 5 packs of 3, cyclically permute each pack:
faro_gen(3, 5, p1=cyc_len(3), interleave=FALSE)

## 15 cards, split into 5 packs of 3, permute the packs as (13542):
print_word(faro_gen(3, 5, p2=as.cycle("(13542)"), interleave=FALSE))


sapply(seq_len(10),function(n){permorder(faro(n,FALSE))}) # OEIS  A002326

plot(as.vector(as.word(faro(10))),type='b')
plot(as.vector(faro_gen(8,5,p1=cyc_len(8)^2,interleave=FALSE)))

permutations documentation built on April 3, 2025, 7:09 p.m.