As a classical object, the Tamari lattice has many generalizations, including ν-Tamari lattices and parabolic Tamari lattices. In this article, we unify these generalizations in a bijective fashion. We first prove that parabolic Tamari lattices are isomorphic to ν-Tamari lattices for bounce paths ν. We then introduce a new combinatorial object called "left-aligned colorable tree", and show that it provides a bijective bridge between various parabolic Catalan objects and certain nested pairs of Dyck paths. As a consequence, we prove the Steep-Bounce Conjecture using a generalization of the famous zeta map in q, t-Catalan combinatorics. A generalization of the zeta map on parking functions, which arises in the theory of diagonal harmonics, is also obtained as a labeled version of our bijection. 1.2.4. Noncrossing Partitions and Left-Aligned Colorable Trees 13 1.2.5. Dyck Paths and Left-Aligned Colorable Trees

The Tamari lattices and the noncrossing partition lattices are important families of lattices that appear in many seemingly unrelated areas of mathematics, such as group theory, combinatorics, representation theory of the symmetric group, algebraic geometry, and many more. They are also deeply connected on a structural level, since the noncrossing partition lattice can be realized as an alternate way of ordering the ground set of the Tamari lattice.Recently, both the Tamari lattices and the noncrossing partition lattices were generalized to parabolic quotients of the symmetric group. In this article we investigate which structural and enumerative properties survive these generalizations.2010 Mathematics Subject Classification. 06B05 (primary), and 05E15 (secondary). 1 quotients of S n . In the following conjecture, H J n (s, t) is a polynomial that is defined on the order ideals of a particular partial order on the transpositions of S J n , and M J n (s, t) is the generating function of the Möbius function of Alt T J n . The precise definitions follow in Section 6. Conjecture 1.9. Let n > 0 and let J = [a, b] ⊆ [n − 1], and let r = n + a − b − 2. The following identity holds if and only if a and b are such that r ∈ {0, 1, . . . , n − 1}: H J n (s + 1, t + 1) = 1 + (s + 1)t r M J n

We investigate the alternate order on a congruence-uniform lattice L as introduced by N. Reading, which we dub the core label order of L. When L can be realized as a poset of regions of a simplicial hyperplane arrangement, the core label order is always a lattice. For general L, however, this fails. We provide an equivalent characterization for the core label order to be a lattice. As a consequence we show that the property of the core label order being a lattice is inherited to lattice quotients. We use the core label order to characterize the congruence-uniform lattices that are Boolean lattices, and we investigate the connection between congruence-uniform lattices whose core label orders are lattices and congruence-uniform lattices of biclosed sets.2010 Mathematics Subject Classification. 06B05 (primary), and 06A07 (secondary).

For an arbitrary Coxeter group W , David Speyer and Nathan Reading defined Cambrian semilattices Cγ as semilattice quotients of the weak order on W induced by certain semilattice homomorphisms. In this article, we define an edge-labeling using the realization of Cambrian semilattices in terms of γ-sortable elements, and show that this is an EL-labeling for every closed interval of Cγ . In addition, we use our labeling to show that every finite open interval in a Cambrian semilattice is either contractible or spherical, and we characterize the spherical intervals, generalizing a result by Nathan Reading.

Abstract. In this article we prove that the lattice of noncrossing partitions is EL-shellable when associated with the well-generated complex reflection group of type G (d, d, n), for d, n ≥ 3, or with the exceptional well-generated complex reflection groups which are no real reflection groups. This result was previously established for the real reflection groups and it can be extended to the well-generated complex reflection group of type G(d, 1, n), for d, n ≥ 3, as well as to three exceptional groups, namely G 25 , G 26 and G 32 , using a braid group argument. We thus conclude that the lattice of noncrossing partitions of any well-generated complex reflection group is EL-shellable. Using this result and a construction by Armstrong and Thomas, we conclude further that the poset of mdivisible noncrossing partitions is EL-shellable for every well-generated complex reflection group. Finally, we derive results on the Möbius function of these posets previously conjectured by Armstrong, Krattenthaler and Tomie.