  Define $S_n^k(\alpha)$ to be the set of permutations of $\{1,2,...,n\}$ with
exactly $k$ fixed points which avoid the pattern $\alpha \in S_m$. Let
$s_n^k(\alpha)$ be the size of $S_n^k(\alpha)$. We investigate $S_n^0(\alpha)$
for all $\alpha \in S_3$ as well as show that
$s_n^k(132)=s_n^k(213)=s_n^k(321)$ and $s_n^k(231)=s_n^k(312)$ for all $0 \leq
k \leq n$.
