  Using the Berele/Remmel/Kerov/Vershik variation of the
Robinson-Schensted-Knuth correspondence, we study the cycle and increasing
subsequence structure after various methods of shuffling. One consequence is a
cycle index for shuffles like: cut a deck into roughly two equal piles,
thoroughly mix the first pile and then riffle it with the second pile.
Conclusions are drawn concerning the distribution of fixed points and the
asymptotic distribution of cycle structure. An upper bound on the convergence
rate is given. Connections are made with extended Schur functions and with work
of Baik and Rains.
