Symmetric Schröder paths and restricted involutions

Let Ak be the set of permutations in the symmetric group Sk with prefix 12. This paper concerns the enumeration of involutions which avoid the set of patterns Ak. We present a bijection between symmetric Schröder paths of length 2n and involutions of length n + 1 avoiding A4. 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 ≥ 3 we determine the generating function for the number of involutions avoiding the subsequences in Ak, according to length, first entry and number of fixed points.