Symmetric Schröder paths and restricted involutions

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 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 A_k, according to length, first entry and number of fixed points.