Centrosymmetric Permutations and Involutions Avoiding 1243 and 2143

TL;DR

This paper characterizes and enumerates centrosymmetric permutations and involutions avoiding the patterns 1243 and 2143, revealing connections to Schröder paths and Pell numbers.

Contribution

It provides a novel characterization and enumeration of centrosymmetric pattern-avoiding permutations and involutions, linking them to Schröder paths and Pell numbers.

Findings

Centrosymmetric permutations avoiding 1243 and 2143 are characterized via reverse-complement invariance.

The enumeration of these permutations is achieved using bijections to Schröder paths.

Centrosymmetric involutions avoiding these patterns are enumerated by Pell numbers.

Abstract

A centrosymmetric permutation is one which is invariant under the reverse-complement operation, or equivalently one whose associated standard Young tableaux under the Robinson-Schensted algorithm are both invariant under the Schutzenberger involution. In this paper, we characterize the set of permutations avoiding 1243 and 2143 whose images under the reverse-complement mapping also avoid these patterns. We also characterize in a simple manner the corresponding Schroder paths under a bijection of Egge and Mansour. We then use these results to enumerate centrosymmetric permutations avoiding the patterns 1243 and 2143. In a similar manner, centrosymmetric involutions avoiding these same patterns are shown to be enumerated by the Pell numbers.