  Type A affine shuffles are compared with riffle shuffles followed by a cut.
Although these probability measures on the symmetric group S_n are different,
they both satisfy a convolution property. Strong evidence is given that when
the underlying parameter $q$ satisfies $gcd(n,q-1)=1$, the induced measures on
conjugacy classes of the symmetric group coincide. This gives rise to
interesting combinatorics concerning the modular equidistribution by major
index of permutations in a given conjugacy class and with a given number of
cyclic descents. It is proved that the use of cuts does not speed up the
convergence rate of riffle shuffles to randomness. Generating functions for the
first pile size in patience sorting from decks with repeated values are
derived. This relates to random matrices.
