  Using symmetric function theory, we study the cycle structure and increasing
subsequence structure of permutations after various shuffling methods,
emphasizing the role of Cauchy type identities and the Robinson-Schensted-Knuth
correspondence. One consequence is a cycle index for riffle shuffles when one
deals from the bottom of the deck; conclusions are drawn concerning the
distribution of fixed points and the asymptotic distribution of cycle
structure. In studying cycle structure of unimodal permutations and their
generalizations we are led to an interesting variation of the RSK
correspondence. This yields a probabilistic interpretation of a Schur-type
function.
