Product decompositions of the symmetric group induced by separable permutations

Abstract

In this paper we consider the rank generating function of a separable permutation π in the weak Bruhat order on the two intervals [id,π] and [π,w0], where w0=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 [id,π] and [π,w0], leading to the rank-symmetry and unimodality of the two graded posets.