Symmetric Schroder paths and restricted involutions

TL;DR

This paper explores the enumeration of involutions avoiding certain patterns, establishing bijections with symmetric Schroder paths and deriving generating functions for various pattern-avoiding involutions.

Contribution

It introduces a bijection between symmetric Schroder paths and involutions avoiding specific patterns, and provides generating functions for these involutions.

Findings

Bijection between symmetric Schroder paths and pattern-avoiding involutions

Explicit generating functions for involutions avoiding A_k patterns

Statistics like fixed points correspond to path steps

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.