Product decompositions of the symmetric group induced by separable permutations

In this paper we consider the rank generating function of a separable permutation ππ in the weak Bruhat order on the two intervals [id,π] and [π,w0][π,w0], where w0=n,n−1,…,1w0=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][π,w0], leading to the rank-symmetry and unimodality of the two graded posets.

► Under weak Bruhat order, separable permutation in SnSn induces two intervals.
► The generating function of SnSn is a product of generating functions in the two intervals.
► There is a rank-preserving bijection between the product of the two intervals and SnSn.
► Explicit formulas for generating functions in the two intervals can be obtained.