On pattern avoidance in matchings and involutions

TL;DR

This paper explores pattern avoidance in involutions and fixed-point-free involutions, providing algorithms, characterizations, and applications to geometric properties of varieties, along with new enumerative results.

Contribution

It introduces an effective algorithm linking two notions of pattern avoidance in involutions and proves conjectures related to geometric smoothness.

Findings

Effective algorithm for pattern avoidance translation

Characterization of involution families where the notions coincide

Proof of McGovern's conjectures on smoothness

Abstract

We study the relationship between two notions of pattern avoidance for involutions in the symmetric group and their restriction to fixed-point-free involutions. The first is classical, while the second appears in the geometry of certain spherical varieties and generalizes the notion of pattern avoidance for perfect matchings studied by Jel\'inek. The first notion can always be expressed in terms of the second, and we give an effective algorithm to do so. We also give partial results characterizing the families of involutions where the converse holds. As a consequence, we prove two conjectures of McGovern characterizing (rational) smoothness of certain varieties. We also give new enumerative results, and conclude by proposing several lines of inquiry that extend our current work.