Perfect Shuffles and Affine Groups
نویسنده
چکیده
For positive integers k and n the group of perfect k-shuffles with a deck of kn cards is a subgroup of the symmetric group Skn. The structure of these groups was found for k = 2 by Diaconis, Graham, and Kantor and for k ≥ 3 and a deck of km cards by Medvedoff and Morrison. They also conjectured that for k = 4 and deck size 2m, m odd, the group is isomorphic to the group of affine transformations of an m-dimensional vector space over the field of order 2. That conjecture is proved in this paper, and a complete conjecture is stated for the the structure of the shuffle groups for all k and n.
منابع مشابه
O ct 1 99 9 Affine shuffles , shuffles with cuts , and patience sorting
Type A affine shuffles are compared with riffle shuffles followed by a cut. Although these probability measures on the symmetric group Sn are different, they both satisfy a convolution property. Strong evidence is given that when the underlying parameter q satisfies gcd(n, q−1) = 1, the induced measures on conjugacy classes of the symmetric group coincide. This gives rise to interesting combina...
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