Exchange Shuffle

A shuffle of a deck of cards obtained by successively exchanging the cards in position 1, 2, ...,  with cards in randomly chosen positions. For , the most frequent permutation is , where  if  is even and either  or  if  is odd (Goldstine and Moews 2000). Amazingly, for  cards, the identity permutation (i.e., the original state before the cards were shuffled) is the most likely (Goldstein and Moews 2000).

REFERENCES:

Goldstein, D. and Moews, D. "The Identity Is the Most Likely Exchange Shuffle for Large ." 6 Oct 2000. https://arxiv.org/abs/math.CO/0010066.

Robbins, D. P. and Bolker, E. D. "The Bias of Three Pseudo-Random Shuffles." Aeq. Math 22, 268-292, 1981.

Schmidt, F. and Simion, R. "Card Shuffling and a Transformation on ." Aeq. Math 44, 11-34, 1992.