# Symmetric Schroder paths and restricted involutions - Mathematics > Combinatorics

Abstract: Let $A k$ be the set of permutations in the symmetric group $S k$ with prefix12. This paper concerns the enumeration of involutions which avoid the set ofpatterns $A k$. We present a bijection between symmetric Schroder paths oflength $2n$ and involutions of length $n+1$ avoiding $\mathcal{A} 4$.Statistics such as the number of right-to-left maxima and fixed points of theinvolution correspond to the number of steps in the symmetric Schroder path ofa particular type. For each $k> 2$ we determine the generating function for thenumber of involutions avoiding the subsequences in $A k$, according to length,first entry and number of fixed points.

Author: Eva Y. P. Deng, Mark Dukes, Toufik Mansour, Susan Y. J. Wu

Source: https://arxiv.org/