A Consecutive Lehmer Code for Parabolic Quotients of the Symmetric Group

TL;DR

This paper introduces a new encoding for parabolic permutations that captures the structure of the parabolic Tamari lattice, providing a simple proof of its isomorphism to a $ u$-Tamari lattice and relating it to existing bijections.

Contribution

It defines a novel encoding for parabolic permutations, proves the lattice isomorphism with a $ u$-Tamari lattice, and clarifies the relationship with a previously used bijection.

Findings

The encoding distinguishes parabolic 231-avoiding permutations.

The componentwise order on codes realizes the parabolic Tamari lattice.

The bijection is related to the map $ heta$ from prior work.

Abstract

In this article we define an encoding for parabolic permutations that distinguishes between parabolic $231$-avoiding permutations. We prove that the componentwise order on these codes realizes the parabolic Tamari lattice, and conclude a direct and simple proof that the parabolic Tamari lattice is isomorphic to a certain $\nu$-Tamari lattice, with an explicit bijection. Furthermore, we prove that this bijection is closely related to the map $\Theta$ used when the lattice isomorphism was first proved in (Ceballos, Fang and M\"uhle, 2020), settling an open problem therein.