Tamari Lattices for Parabolic Quotients of the Symmetric Group

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.