Restricted involutions and Motzkin paths

TL;DR

This paper explores bijections between involutions avoiding specific patterns and Motzkin paths, providing new characterizations and insights into restricted involution subsets using combinatorial bijections.

Contribution

It introduces new bijections linking restricted involutions to Motzkin paths, extending understanding of pattern-avoiding involutions and their combinatorial structures.

Findings

Bijections between involutions avoiding 4321/3412 and Motzkin paths

Characterizations of Motzkin paths for pattern-avoiding involutions

Analysis of fixed point free and centrosymmetric involutions

Abstract

We show how a bijection due to Biane between involutions and labelled Motzkin paths yields bijections between Motzkin paths and two families of restricted involutions that are counted by Motzkin numbers, namely, involutions avoiding 4321 and 3412. As a consequence, we derive characterizations of Motzkin paths corresponding to involutions avoiding either 4321 or 3412 together with any pattern of length 3. Furthermore, we exploit the described bijection to study some notable subsets of the set of restricted involutions, namely, fixed point free and centrosymmetric restricted involutions.