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This paper finds and analyzes a formula for the total variation distance between iterations of rie shues and iterations of "cut and then rie shue". This allows one to obtain information about the convergence rate of permutation statistics (such as the length of the longest increasing subsequence or position of a given card) under these processes. Similar results are given for ane shues, which allow us to determine their convergence rate to randomness.

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