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We evaluate the probabilities of various events under the uniformdistribution on the set of 312-avoiding permutations of 1,...,N. We deriveexact formulas for the probability that the ith element of a random permutationis a specific value less than i, and for joint probabilities of two suchevents. In addition, we obtain asymptotic approximations to these probabilitiesfor large N when the elements are not close to the boundaries or to each other.We also evaluate the probability that the graph of a random 312-avoidingpermutation has k specified decreasing points, and we show that for large N thepoints below the diagonal look like trajectories of a random walk.

Structure of Random 312-Avoiding Permutations