riffle {partitions}

Riffle shuffles

Description

Enumeration of riffle shuffles

Usage

genrif(v)
riffle(p,q=p)

Arguments

p, q, v

In function riffle(), integers p,q specify the length of the two increasing sequences. In function genrif(), the elements of v specify the lengths of all the increasing sequences: there are sum(v) cards in the pack

Details

A riffle shuffle is a permutation of integers \(1,2,\ldots,n\) containing one or two rising sequences.

A generalized riffle shuffle, or \(r\)-riffle shuffle, contains at most \(r\) rising sequences. This is not implemented in the package (earlier versions included a buggy version; the difficulty is ensuring that sequences do not appear more than once).

• riffle(p,q) returns all riffle shuffles with rising sequences of 1:p and (p+1):q

• genrif(v) returns all riffle shuffles with rising sequences having lengths the entries of v, the deck being numbered consecutively

Value

Returns a matrix of class partition with columns being riffle shuffles

Note

When we say “contains \(r\) rising sequences” we generally mean “contains at most \(r\) rising sequences”

Author(s)

Robin K. S. Hankin

See Also

parts

Examples

riffle(3, 4)
                                                                          
##: [1,] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
##: [2,] 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 1 1 1 1 1 1 1 1 1 1 5 5 5 5 5 5 5 5 5 5
##: [3,] 3 4 4 4 4 2 2 2 2 5 5 5 5 5 5 2 2 2 2 5 5 5 5 5 5 1 1 1 1 1 1 6 6 6 6
##: [4,] 4 3 5 5 5 3 5 5 5 2 2 2 6 6 6 3 5 5 5 2 2 2 6 6 6 2 2 2 6 6 6 1 1 1 7
##: [5,] 5 5 3 6 6 5 3 6 6 3 6 6 2 2 7 5 3 6 6 3 6 6 2 2 7 3 6 6 2 2 7 2 2 7 1
##: [6,] 6 6 6 3 7 6 6 3 7 6 3 7 3 7 2 6 6 3 7 6 3 7 3 7 2 6 3 7 3 7 2 3 7 2 2
##: [7,] 7 7 7 7 3 7 7 7 3 7 7 3 7 3 3 7 7 7 3 7 7 3 7 3 3 7 7 3 7 3 3 7 3 3 3
genrif(1:3)
                                                                              
##: [1,] 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4
##: [2,] 2 2 2 2 4 4 4 4 4 4 1 1 1 1 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 1 1 1 1 1 1 2
##: [3,] 3 4 4 4 2 2 2 5 5 5 3 4 4 4 1 4 4 4 1 1 1 3 3 3 5 5 5 5 5 5 2 2 2 5 5 5 1
##: [4,] 4 3 5 5 3 5 5 2 2 6 4 3 5 5 4 1 5 5 3 5 5 1 5 5 1 1 3 3 6 6 3 5 5 2 2 6 3
##: [5,] 5 5 3 6 5 3 6 3 6 2 5 5 3 6 5 5 1 6 5 3 6 5 1 6 3 6 1 6 1 3 5 3 6 3 6 2 5
##: [6,] 6 6 6 3 6 6 3 6 3 3 6 6 6 3 6 6 6 1 6 6 3 6 6 1 6 3 6 1 3 1 6 6 3 6 3 3 6
                                                  
##: [1,] 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
##: [2,] 2 2 2 2 2 2 2 2 2 2 2 5 5 5 5 5 5 5 5 5 5 5 5
##: [3,] 1 1 3 3 3 5 5 5 5 5 5 1 1 1 2 2 2 2 2 2 6 6 6
##: [4,] 5 5 1 5 5 1 1 3 3 6 6 2 2 6 1 1 3 3 6 6 1 2 2
##: [5,] 3 6 5 1 6 3 6 1 6 1 3 3 6 2 3 6 1 6 1 3 2 1 3
##: [6,] 6 3 6 6 1 6 3 6 1 3 1 6 3 3 6 3 6 1 3 1 3 3 1