Permutations Restricted by Two Distinct Patterns of Length Three

Define Sn(R;T ) to be the number of permutations on n letters which avoid all patterns in the set R and contain each pattern in the multiset T exactly once. In this paper we enumerate Sn({α}; {β}) and Sn(∅; {α, β}) for all α 6= β ∈ S3. We show that there are five Wilf-like classes associated with each of Sn({α}; {β}) and Sn(∅; {α, β}) for all α 6= β ∈ S3.