Pattern avoidance and the Bruhat order on involutions

TL;DR

This paper characterizes when the principal order ideal in the Bruhat order on involutions forms a Boolean lattice, based on pattern avoidance, and explores related enumerations and bijections with Motzkin paths.

Contribution

It provides a pattern avoidance criterion for Boolean lattice structures in Bruhat order on involutions and extends this to signed permutations with enumerative results and combinatorial bijections.

Findings

Boolean lattice occurs iff involution avoids patterns 4321, 45312, 456123

Enumerates involutions with this property using natural statistics

Establishes a bijection with Motzkin paths

Abstract

We show that the principal order ideal below an element w in the Bruhat order on involutions in a symmetric group is a Boolean lattice if and only if w avoids the patterns 4321, 45312 and 456123. Similar criteria for signed permutations are also stated. Involutions with this property are enumerated with respect to natural statistics. In this context, a bijective correspondence with certain Motzkin paths is demonstrated.