Patterns in Random Permutations Avoiding Some Sets of Multiple Patterns

We consider a random permutation drawn from the set of permutations of length n that avoid some given set of patterns of length 3. We show that the number of occurrences of another pattern σ has a limit distribution, after suitable scaling. In several cases, the number is asymptotically normal; this contrasts to the cases of permutations avoiding a single pattern of length 3 studied in earlier papers.