Odd diagrams, Bruhat order, and pattern avoidance

TL;DR

This paper investigates the structure of permutations with identical odd diagrams, establishing their connection to pattern avoidance and Bruhat intervals, and provides explicit descriptions of Bruhat edges within these classes.

Contribution

It proves a conjecture linking odd diagram classes to 213- and 312-avoiding permutations and shows each class forms a Bruhat interval with explicit edge descriptions.

Findings

Odd diagram classes correspond to certain pattern-avoiding permutations.

Each odd diagram class forms a Bruhat interval.

Explicit Bruhat edge descriptions within classes are provided.

Abstract

The odd diagram of a permutation is a subset of the classical diagram with additional parity conditions. In this paper, we study classes of permutations with the same odd diagram, which we call odd diagram classes. First, we prove a conjecture relating odd diagram classes and 213- and 312-avoiding permutations. Secondly, we show that each odd diagram class is a Bruhat interval. Instrumental to our proofs is an explicit description of the Bruhat edges that link permutations in a class.