A refinement of weak order intervals into distributive lattices

TL;DR

This paper demonstrates that intervals in the left weak order on symmetric groups can be refined into distributive lattices using Lehmer codes, revealing new structural insights and symmetric properties.

Contribution

It introduces a method to represent weak order intervals as distributive lattices via Lehmer codes and constructs a related poset with isomorphic lattice of order ideals.

Findings

Lehmer codes form a distributive lattice under product order.

The rank-generating function matches that of the original interval.

At least (⌊n/2⌋)! permutations form rank-symmetric intervals.

Abstract

In this paper we consider arbitrary intervals in the left weak order on the symmetric group $S_n$. 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 $S_n$ that form a rank-symmetric interval in the weak order.