Symmetric Schrِder 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 2 n and involutions of length n + 1 avoiding 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 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.