Tamari Lattices for Parabolic Quotients of the Symmetric Group

TL;DR

This paper extends the Tamari lattice to parabolic quotients of symmetric groups and Coxeter groups, establishing bijections among permutations, noncrossing, and nonnesting partitions, and introduces a generalized aligned inversion set condition.

Contribution

It introduces a generalized Tamari lattice framework for parabolic quotients of Coxeter groups, unifying various combinatorial objects through a new aligned inversion set condition.

Findings

Bijection between $231$-avoiding permutations and set partitions

Extension of constructions to all finite Coxeter groups

Introduction of a generalized aligned inversion set condition

Abstract

We generalize the Tamari lattice by extending the notions of $231$-avoiding permutations, noncrossing set partitions, and nonnesting set partitions to parabolic quotients of the symmetric group $\mathfrak{S}_{n}$. We show bijectively that these three objects are equinumerous. We show how to extend these constructions to parabolic quotients of any finite Coxeter group. The main ingredient is a certain aligned condition of inversion sets; a concept which can in fact be generalized to any reduced expression of any element in any (not necessarily finite) Coxeter group.