The Weak Bruhat Order and Separable Permutations

TL;DR

This paper studies the rank generating functions of separable permutations within the weak Bruhat order, revealing a surprising product relation and establishing properties like rank-symmetry and unimodality of related posets.

Contribution

It introduces explicit formulas for rank generating functions of separable permutations in the weak Bruhat order and uncovers a novel product relation between these functions.

Findings

Product of generating functions equals the symmetric group's generating function.

Explicit formulas for rank generating functions are derived.

The posets are shown to be rank-symmetric and unimodal.

Abstract

In this paper we consider the rank generating function of a separable permutation $\pi$ in the weak Bruhat order on the two intervals $[\text{id}, \pi]$ and $[\pi, w_0]$, where $w_0 = n,(n-1),..., 1$. We show a surprising result that the product of these two generating functions is the generating function for the symmetric group with the weak order. We then obtain explicit formulas for the rank generating functions on $[\text{id}, \pi]$ and $[\pi, w_0]$, which leads to the rank-symmetry and unimodality of the two graded posets.