The absence of a pattern and the number of occurrences of another

Following a question of J. Cooper, we study the expected number of occurrences of a given permutation pattern $q$ in permutations that avoid another given pattern $r$. In some cases, we find the pattern that occurs least often, (resp. most often) in all $r$-avoiding permutations. We also prove a few exact enumeration formulae, some of which are surprising.