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Kirk Lecture: The mathematics of Shuffling

Presented by: 
Cheryl Praeger
Tuesday 17th March 2020 - 16:00 to 17:00
INI Seminar Room 1
The crux of a card trick performed with a deck of cards usually depends on understanding how shuffles of the deck change the order of the cards. By understanding which permutations are possible, one knows if a given card may be brought into a certain position. The mathematics of shuffling a deck of 2n cards with two ``perfect shuffles'' was studied thoroughly by Diaconis, Graham and Kantor in 1983. I will report on our efforts to understand 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! possible shuffles.

A conjecture of Medvedoff and Morrison suggests that all possible permutations of the deck of cards are achieved, as long as k is not 4 and n is not a power of k. We confirm this conjecture for three doubly infinite families of integers, including all (k, n) with k > n. We initiate a more general study of shuffle groups, which admit an arbitrary subgroup of shuffles. This is joint work with Carmen Amarra and Luke Morgan.
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons