The fine structure of 321 avoiding involutions

TL;DR

This paper analyzes involutions avoiding the pattern 321, deriving algebraic generating functions for various subclasses, and provides combinatorial interpretations using Motzkin and Dyck paths.

Contribution

It computes explicit algebraic generating functions for involutions avoiding 321 and their subclasses, and offers combinatorial interpretations via Motzkin and Dyck paths.

Findings

Algebraic generating functions for involutions avoiding 321 are derived.

Simple involutions avoiding 321 are counted by Riordan's numbers.

Combinatorial interpretations are provided through Motzkin and Dyck paths.

Abstract

We study the involutions belonging to the class of 321 avoiding permutations. We calculate the algebraic generating functions of the set containing the involutions avoiding 321 and of some of its subsets. Precisely we determine the algebraic generating functions of the involutions that are expansions of 12, of those expansions of 21, of the simple ones and of their expansions. The graphics of the simple involutions are caracterized. Being the simple involutions avoiding 321 counted by Riordan's numbers, a combinatoric interpretation of the results is illustrated through a class of Motzkin paths. Another interpretation is given through Dyck paths.