Wachs permutations, Bruhat order and weak order

TL;DR

This paper investigates the structure of Wachs and signed Wachs permutations under Bruhat and weak orders, revealing their grading, rank functions, and explicit structures, advancing understanding of these permutation classes.

Contribution

It characterizes the ordering, covering relations, and computes characteristic polynomials for Wachs permutations under Bruhat order, and explicitly describes their structure under weak order.

Findings

Orders are graded with determined rank functions

Explicit structure of Wachs permutations under weak order

Characteristic polynomials computed for Bruhat order

Abstract

We study the partial orders induced on Wachs and signed Wachs permutations by the Bruhat and weak orders of the symmetric and hyperoctahedral groups. We show that these orders are graded, determine their rank function, characterize their ordering and covering relations, and compute their characteristic polynomials, when partially ordered by Bruhat order, and determine their structure explicitly when partially ordered by right weak order.