Symmetric Schröder paths and restricted involutions

Let AkAk be the set of permutations in the symmetric group SkSk with prefix 12. This paper concerns the enumeration of involutions which avoid the set of patterns AkAk. We present a bijection between symmetric Schröder paths of length 2n2n and involutions of length n+1n+1 avoiding A4A4. 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 Schröder path of a particular type. For each k≥3k≥3 we determine the generating function for the number of involutions avoiding the subsequences in AkAk, according to length, first entry and number of fixed points.