A Refinement of Weak Order Intervals into Distributive Lattices

Abstract

In this paper we consider arbitrary intervals in the left weak order on the symmetric group Sn. We show that the Lehmer codes of permutations in an interval form a distributive lattice under the product order. Furthermore, the rank-generating function of this distributive lattice matches that of the weak order interval. We construct a poset such that its lattice of order ideals is isomorphic to the lattice of Lehmer codes of permutations in the given interval. We show that there are at least \({\left(\lfloor {\frac{n}{2}} \rfloor \right)!}\) permutations in Sn that form a rank-symmetric interval in the weak order.