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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 socalled “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 ⩾ 2, ℓ^{e} > 4, f not a multiple of e}; and all (k, n) with k = 2^{e} ⩾ 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.
Original language  English 

Pages (fromto)  807856 
Number of pages  50 
Journal  Israel Journal of Mathematics 
Volume  244 
Issue number  2 
Early online date  9 Sept 2021 
DOIs  
Publication status  Published  Sept 2021 
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Dive into the research topics of 'Generalised shuffle groups'. Together they form a unique fingerprint.Projects
 1 Finished

Structure theory for permutation groups and local graph theory conjectures
ARC Australian Research Council
1/01/16 → 31/01/19
Project: Research