  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.
