A Card Shuffling Analysis of Deformations of the Plancherel Measure of the Symmetric Group

We study deformations of the Plancherel measure of the symmetric group by lifting them to the symmetric group and using combinatorics of card shuffling. The existing methods for analyzing deformations of Plancherel measure are not obviously applicable to the examples in this paper. The main idea of this paper is to find and analyze a formula for the total variation distance between iterations of riffle shuffles and iterations of "cut and then riffle shuffle". Similar results are given for affine shuffles, which allow us to determine their convergence rate to randomness

A rule of thumb for riffle shuffling

We study how many riffle shuffles are required to mix n cards if only certain features of the deck are of interest, e.g. suits disregarded or only the colors of interest. For these features, the number of shuffles drops from 3/2 log_2(n) to log_2(n). We derive closed formulae and an asymptotic `rule of thumb' formula which is remarkably accurate.Comment: 27 pages, 5 table