ON SAMPLING FROM AN URN, WITHOUT REPLACEMENT

Put n balls, with weights w(1),...,w(n) into an urn. Draw them out, one at a time, each time picking a ball with chance its weight, relative to what's left. This generates a random permutation and one can ask 'what does it 'look like'. The model is used by psychologists, to settle hands in poker tournaments and for an actual  winning strategy when betting on horse races. In joint work with Jacopo Borga,  Sourav Chatterjee and Gene Kim, we can 'do the combinatorics'. There are limiting objects called 'permutons' (like graphons-but for permutations), and the local pattern structure has its own, different story. There are lots of open problems; for example, we don't understand the behaviour of the number of fixed points (or cycle structure).