Stephen Connor (University of York) – Cutoff for a one-sided transposition shuffle

Consider the following method for shuffling a deck of cards: at each step we select a card uniformly at random, and then transpose this with a card chosen uniformly from beneath the selected card. We call this a "one-sided transposition shuffle", and ask the question: how many shuffles does it take to "randomise" the deck? It turns out that we can give a precise answer to this question using a combination of ideas from probability and representation theory of the symmetric group; in particular, we are able to deduce an explicit formula for the eigenvalues of the shuffle by relating them to Young tableaux.

Joint work with O. Matheau-Raven and M. Bate.