  Let $A_k$ be the set of permutations in the symmetric group $S_k$ with prefix
12. This paper concerns the enumeration of involutions which avoid the set of
patterns $A_k$. We present a bijection between symmetric Schroder paths of
length $2n$ and involutions of length $n+1$ avoiding $\mathcal{A}_4$.
Statistics such as the number of right-to-left maxima and fixed points of the
involution correspond to the number of steps in the symmetric Schroder path of
a particular type. For each $k> 2$ we determine the generating function for the
number of involutions avoiding the subsequences in $A_k$, according to length,
first entry and number of fixed points.
