The fine structure of the sets of involutions avoiding 4321 or 3412

TL;DR

This paper analyzes the detailed structure of involutions avoiding specific patterns, linking decomposition theorems with labelled Motzkin paths, and provides generating functions for simple involutions in these sets.

Contribution

It introduces a novel connection between involution pattern avoidance and labelled Motzkin paths, and computes generating functions for simple involutions in these classes.

Findings

The algebraic generating function of simple involutions in I(4321) is derived.

The set I(3412) contains no simple involutions of length greater than 2.

Reverse-complement bijection preserves the fine structure of these involution sets.

Abstract

We study the fine structure of the sets of involutions avoiding either 4312 (I(4321)) or 3412 (I(3412)), connecting the point of view of the decomposition theorems with the one of the associated labelled Motzkin paths. The algebraic generating function of the simple involutions in I(4321) is given, together with other generating functions, while the set I(3412) is shown containing no simple involutions of length n>2. The reverse-complement bijection maintains the fine structures of I(4321) and trivially of I(3412).