An Involution on Involutions and a Generalization of Layered Permutations

TL;DR

This paper explores an involution on involutions derived from Young tableaux and generalizes layered permutations, providing new insights into combinatorial structures and their symmetries.

Contribution

It introduces a novel involution on involutions and offers a generalization of layered permutations, some of which can be defined independently of the Robinson-Schensted correspondence.

Findings

Defined an involution on involutions via Young tableaux transposes

Generalized layered permutations beyond existing frameworks

Provided alternative definitions avoiding Robinson-Schensted correspondence

Abstract

Taking transposes of Standard Young Tableaux defines a natural involution on the set $I(n)$ of involutions of length $n$ via the the Robinson-Schensted correspondence. In some cases, this involution can be defined without resorting to the Robinson-Schensted correspondence. As a byproduct, we get an interesting generalization of layered permutations.