Tamari lattices and noncrossing partitions in type B

Abstract

The usual, or type A n , Tamari lattice is a partial order on T A n , the triangulations of an (n + 3)-gon. We define a partial order on T B n , the set of centrally symmetric triangulations of a (2n + 2)-gon. We show that it is a lattice, and that it shares certain other nice properties of the A n Tamari lattice, and therefore that it deserves to be considered the B n Tamari lattice. We also define a bijection between T B n and the noncrossing partitions of type B n defined by Reiner.

On Noncrossing and Nonnesting Partitions of Type D

“…Recently, Victor Reiner [19] and Alexander Postnikov, see Christos Athanasiadis' paper [2, §6, Remark 2] generalized the notion of, respectively, noncrossing and nonnesting partitions to all classical reflection groups, and a new very active research area sprung up, namely generalizing previous known results held for noncrossing and nonnesting partitions of type A to their type B and D analogue, see e.g. [1,3,13,26,27] and the references therein.…”

“…[4,7,8,9,12,14,15,18,21,22,23,24,25] and the references therein.Recently, Victor Reiner [19] and Alexander Postnikov, see Christos Athanasiadis' paper [2, §6, Remark 2] generalized the notion of, respectively, noncrossing and nonnesting partitions to all classical reflection groups, and a new very active research area sprung up, namely generalizing previous known results held for noncrossing and nonnesting partitions of type A to their type B and D analogue, see e.g. [1,3,13,26,27] and the references therein.In particular, it is well known that in type A the number of noncrossing partition equals the number of nonnesting partition, and several bijections have been established.Recently one of us in [16,17] solved the corresponding problem for the type B (and type C) presenting an explicit bijection which falls back to one already known when restricted to the type A.The goal of this paper is to present an explicit bijection for the type D, therefore solving the problem for all classical Weyl groups, and to show that this bijection has interesting additional combinatorial properties. For instance, for any classical reflection group and any partition it is possible to define three different maps, called openers, closers, and transients, going from the partition itself to subsets of the involved reflection group, and we show that our bijection preserves all these functions.…”

Lattices from graph associahedra and subalgebras of the Malvenuto-Reutenauer algebra

“…The graph associahedron P Cn is used to study the self-linking of knots [2] or to tile the moduli space Z n in [8], whereas the Type B n associahedron arises in the theory of cluster algebras [10]. From the Coxeter-Catalan point of view, the vertices of the Type B n associahedron can be partially ordered in several ways, which are called Cambrian lattices; [30], [36]. A Cambrian lattice is a certain lattice quotient of the weak order of a finite Coxeter system.…”

The two higher Stasheff-Tamari orders are equal

“…Generalisations of, and variations on, of the Tamari lattice is a large subject in itself, and includes Tamari lattices in other Dynkin types [Tho06], Cambrian lattices [Rea06], lattices of torsion classes of cluster-tilted algebras [GM19], m-Tamari lattices [BP12; BFP11], ν-Tamari lattices [PV17], Dyck lattices [Knu11;Dis+12], generalised Tamari orders [Ron12], and Grassmann-Tamari orders [SSW17]. However, the higher Stasheff-Tamari orders hold a particularly special position amongst these because, as we have seen, they encode higher-dimensional information hidden in the Tamari lattice itself, rather than being only variations of the Tamari lattice.…”