Refined Restricted Involutions
  Define $I_n^k(\alpha)$ to be the set of involutions of $\{1,2,...,n\}$ with exactly $k$ fixed points which avoid the pattern $\alpha \in S_i$, for some $i \geq 2$, and define $I_n^k(\emptyset;\alpha)$ to be the set of involutions of $\{1,2,...,n\}$ with exactly $k$ fixed points which contain the pattern $\alpha \in S_i$, for some $i \geq 2$, exactly once. Let $i_n^k(\alpha)$ be the number of elements in $I_n^k(\alpha)$ and let $i_n^k(\emptyset;\alpha)$ be the number of elements in $I_n^k(\emptyset;\alpha)$. We investigate $I_n^k(\alpha)$ and $I_n^k(\emptyset;\alpha)$ for all $\alpha \in S_3$. In particular, we show that $i_n^k(132)=i_n^k(213)=i_n^k(321)$, $i_n^k(231)=i_n^k(312)$, $i_n^k(\emptyset;132) =i_n^k(\emptyset;213)$, and $i_n^k(\emptyset;231)=i_n^k(\emptyset;312)$ for all $0 \leq k \leq n$. 
Emeric Deutsch, Aaron Robertson, Dan Saracino