Generalised shuffle groups

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) ∈{(ℓe, ℓf) ∣ ℓ ⩾ 2, ℓe > 4, f not a multiple of e}; and all (k, n) with k = 2e ⩾ 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.