  Recently Wilson [Ann. Appl. Probab. 14 (2004) 274--325] introduced an
important new technique for lower bounding the mixing time of a Markov chain.
In this paper we extend Wilson's technique to find lower bounds of the correct
order for card shuffling Markov chains where at each time step a random card is
picked and put at the top of the deck. Two classes of such shuffles are
addressed, one where the probability that a given card is picked at a given
time step depends on its identity, the so-called move-to-front scheme, and one
where it depends on its position. For the move-to-front scheme, a test function
that is a combination of several different eigenvectors of the transition
matrix is used. A general method for finding and using such a test function,
under a natural negative dependence condition, is introduced. It is shown that
the correct order of the mixing time is given by the biased coupon collector's
problem corresponding to the move-to-front scheme at hand. For the second
class, a version of Wilson's technique for complex-valued
eigenvalues/eigenvectors is used. Such variants were presented in [Random Walks
and Geometry (2004) 515--532] and [Electron. Comm. Probab. 8 (2003) 77--85].
Here we present another such variant which seems to be the most natural one for
this particular class of problems. To find the eigenvalues for the general case
of the second class of problems is difficult, so we restrict attention to two
special cases. In the first case the card that is moved to the top is picked
uniformly at random from the bottom $k=k(n)=o(n)$ cards, and we find the lower
bound $(n^3/(4\pi^2k(k-1)))\log n$. Via a coupling, an upper bound exceeding
this by only a factor 4 is found. This generalizes Wilson's [Electron. Comm.
Probab. 8 (2003) 77--85] result on the Rudvalis shuffle and Goel's [Ann. Appl.
Probab. 16 (2006) 30--55] result on top-to-bottom shuffles. In the second case
the card moved to the top is, with probability 1/2, the bottom card and with
probability 1/2, the card at position $n-k$. Here the lower bound is again of
order $(n^3/k^2)\log n$, but in this case this does not seem to be tight unless
$k=O(1)$. What the correct order of mixing is in this case is an open question.
We show that when $k=n/2$, it is at least $\Theta(n^2)$.
