Luke Morgan: Shuffling cards with groups
Povzetek. There are two standard ways to shuffle a deck of cards, the in and out shuffles. For the in shuffle, divide the deck into two piles, hold one pile in each hand and then perfectly interlace the piles, with the top card from the left hand pile being on top of the resulting stack of cards. For the out shuffle, the top card from the right hand pile ends up on top of the resulting stack. Standard card tricks are based on knowing what permutations of the deck of cards may be achieved just by performing the in and out shuffles.
Mathematicians answer this question by solving the problem of what permutation group is generated by these two shuffles. Diaconis, Graham and Kantor were the first to solve this problem in full generality - for decks of size 2*n. The answer is usually “as big as possible”, but with some rather beautiful and surprising exceptions. In this talk, I’ll explain how the number of permutations is limited, and give some hints about how to obtain different permutations of the deck. I’ll also present a more general question about a “many handed dealer” who shuffles k*n cards divided into k piles.