Affine shuffles, shuffles with cuts, the Whitehouse module, and patience sorting

1999/10/18 by Jason Fulman, Fulman, Jason
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR) #math.CO #math.PR

paper · pdf · doi:10.48550/arxiv.math/9910087

Galley version for J. Alg.; minor revisions in Sec. 3

arxiv created 2000/05/12 · arxiv updated 2009/11/30

Abstract

Type A affine shuffles are compared with riffle shuffles followed by a cut. Although these probability measures on the symmetric group S_n 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 combinatorics concerning the modular equidistribution by major index of permutations in a given conjugacy class and with a given number of cyclic descents. It is proved that the use of cuts does not speed up the convergence rate of riffle shuffles to randomness. Generating functions for the first pile size in patience sorting from decks with repeated values are derived. This relates to random matrices.

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