  In this thesis we introduce a new type of card shuffle called the one-sided
transposition shuffle. At each step a card is chosen uniformly from the pack
and then transposed with another card chosen uniformly from below it. This
defines a random walk on the symmetric group generated by a distribution which
is non-constant on the conjugacy class of transpositions. Nevertheless, we
provide an explicit formula for all eigenvalues of the shuffle by demonstrating
a useful correspondence between eigenvalues and standard Young tableaux. This
allows us to prove the existence of a total-variation cutoff for the one-sided
transposition shuffle at time $n\log n$. We also study weighted generalisations
of the one-sided transposition shuffle called biased one-sided transposition
shuffles. We compute the full spectrum for every biased one-sided transposition
shuffle, and prove the existence of a total variation cutoff for certain
choices of weighted distribution. In particular, we recover the eigenvalues and
well known mixing time of the classical random transposition shuffle. We study
the hyperoctahedral group as an extension of the symmetric group, and formulate
the one-sided transposition shuffle and random transposition shuffle as random
walks on this new group. We determine the spectrum of each hyperoctahedral
shuffle by developing a correspondence between their eigenvalues and standard
Young bi-tableaux. We prove that the one-sided transposition shuffle on the
hyperoctahedral group exhibits a cutoff at $n\log n$, the same time as its
symmetric group counterpart. We conjecture that this results extends to the
biased one-sided transposition shuffles and the random transposition shuffle on
the hyperoctahedral group.
