Cutoff for a one-sided transposition shuffle

Abstract

We introduce a new type of card shuffle called one-sided transpositions. 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 nonconstant 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 $\mathit{n}log\mathit{n}$. We also study a weighted generalisation of the shuffle which, in particular, allows us to recover the well-known mixing time of the classical random transposition shuffle.