  Define $S_n(R;T)$ 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 $S_n(\{\alpha\};\{\beta\})$ and
$S_n(\emptyset;\{\alpha,\beta\})$ for all $\alpha \neq \beta \in S_3$. The
results for $S_n(\{\alpha\};\{\beta\})$ follow from two papers by Mansour and
Vainshtein.
