Symmetric Schroder paths and restricted involutions

Abstract

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.

11 pages