Card shuffling and a transformation on S _n

We consider two card shuffling schemes. The first, which has appeared in the literature previously ([G], [RB], [T]), is as follows: start with a deck ofn cards, and pick a random tuplet ∈ { 1, 2, ⋯, n} ⁿ; interchange cards 1 andt ₁, then interchange cards 2 andt ₂, etc. The second scheme, which can be viewed as a transformation on the symmetric groupS _n , is given by the restriction of the former shuffling scheme to tuplest which form a permutation of {1, 2,⋯,n}.

We determine the bias of each of these shuffling schemes with respect to the sets of transpositions and derangements, and the expected number of fixed points of a permutation generated by each of these shuffling schemes. For the latter scheme we prove combinatorially that the permutation which arises with the highest probability is the identity. The same question is open for the former scheme. We refute a candidate answer suggested by numerical evidence [RB].

Authors and Affiliations

1. Department of Mathematics, Bryn Mawr College, 19010, Bryn Mawr, PA, USA

   Frank Schmidt & Rodica Simion

2. Deparment of Mathematics, The George Washington University, 20052, Washington, D.C., USA

   Frank Schmidt & Rodica Simion

This work was carried out in part while R.S. was visiting the Institute for Mathematics and its Applications and was partly supported through NSF Grant CCR-8707539.

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