Inspired by real-world applications such as the assignment of pupils to schools or allocation social housing, one-sided matching problem studies how a set agents can be assigned objects when have preferences over objects, but not vice versa. For fairness reasons, most mechanisms use randomness, and therefore result in probabilistic assignment. We study decomposing these assignments into weighte...

We introduce a new type of card shuffle called one-sided transpositions. At each step is chosen uniformly from the pack and then transposed with another below it. This defines random walk on symmetric group generated by distribution which nonconstant conjugacy class Nevertheless, we provide an explicit formula for all eigenvalues demonstrating useful correspondence between standard Young tablea...