Generalised shuffle groups

2019/08/14 by Amarra, Carmen, Morgan, Luke, Praeger, Cheryl E.
#05E18 #20B25 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1908.05128

Abstract

The mathematics of shuffling a deck of 2n cards with two "perfect shuffles" was brought into clarity by Diaconis, Graham and Kantor. Here we consider a generalisation of this problem, with a so-called "many handed dealer" shuffling kn cards by cutting into k piles with n cards in each pile and using k! shuffles. A conjecture of Medvedoff and Morrison suggests that all possible permutations of the deck of cards are achieved, so long as k≠ 4 and n is not a power of k. We confirm this conjecture for three doubly infinite families of integers: all (k,n) with k>n; all (k, n)∈ \ (ℓ^e, ℓ^f )| ℓ \geqslant 2, ℓ^e>4, f not a multiple of e\; and all (k,n) with k=2^e\geqslant 4 and n not a power of 2. We open up a more general study of shuffle groups, which admit an arbitrary subgroup of shuffles.

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