In this paper we consider arbitrary intervals in the left weak order on the symmetric group $S_n$. We show that the Lehmer codes of permutations in an interval form a distributive lattice under the product order. Furthermore, the rank-generating function of this distributive lattice matches that of the weak order interval. We construct a poset such that its lattice of order ideals is isomorphic to the lattice of Lehmer codes of permutations in the given interval. We show that there are at least $\left(\lfloor\frac{n}{2}\rfloor\right)!$ permutations in $S_n$ that form a rank-symmetric interval in the weak order.
This paper proves a dual version of a theorem of Oystein Ore for every distributive interval of finite groups [H,G] of index |G:H|<9720, and for every boolean interval of rank <7. It has applications to representation theory for every finite group.
This paper gives a self-contained group-theoretic proof of a dual version of a theorem of Ore on distributive intervals of finite groups. We deduce a bridge between combinatorics and representations in finite group theory.
A simple but elegant result of Rival states that every sublattice $L$ of a finite distributive lattice $\mathcal{P}$ can be constructed from $\mathcal{P}$ by removing a particular family $\mathcal{I}_L$ of its irreducible intervals. Applying this in the case that $\mathcal{P}$ is a product of a finite set $\mathcal{C}$ of chains, we get a one-to-one correspondence $L \mapsto \mathcal{D}_\mathcal{P}(L)$ between the sublattices of $\mathcal{P}$ and the preorders spanned by a canonical sublattice $\mathcal{C}^\infty$ of $\mathcal{P}$. We then show that $L$ is a tight sublattice of the product of …
We study the appearance of notable interval structures -- lattices, modular lattices, distributive lattices, and boolean lattices -- in both the Bruhat and weak orders of Coxeter groups. We collect and expand upon known results for principal order ideals, including pattern characterizations and enumerations for the symmetric group. This segues naturally into a similar analysis for arbitrary intervals, although the results are less characterizing for the Bruhat order at this generality. In counterpoint, however, we obtain a full characterization for intervals starting at rank one in the symmetr…
We introduce a new class of poset edge labelings for locally finite lattices which we call $SB$-labelings. We prove for finite lattices which admit an $SB$-labeling that each open interval has the homotopy type of a ball or of a sphere of some dimension. Natural examples include the weak order, the Tamari lattice, and the finite distributive lattices.
In this paper, we define a property, trimness, for lattices. Trimness is a not-necessarily-graded generalization of distributivity; in particular, if a lattice is trim and graded, it is distributive. Trimness is preserved under taking intervals and suitable sublattices. Trim lattices satisfy a weakened form of modularity. The order complex of a trim lattice is contractible or homotopic to a sphere; the latter holds exactly if the maximum element of the lattice is a join of atoms. Other than distributive lattices, the main examples of trim lattices are the Tamari lattices and various generaliza…
Let A be a finite set with at least two elements. The composition of two classes I and J of operations on A, is defined as the set of all compositions of functions in I with functions in J. This binary operation gives a monoid structure to the set E_A of all equational classes of operations on A. The set E_A of equational classes of operations on A also constitutes a complete distributive lattice under intersection and union. Clones of operations, i.e. classes containing all projections and idempotent under class composition, also form a lattice which is strictly contained in E_A. In the Boole…
We call a lattice crosscut-simplicial if the crosscut complex of every atomic interval is equal to the boundary of a simplex. Every interval of such a lattice is either contractible or homotopy equivalent to a sphere. Recently, Hersh and Meszaros introduced SB-labellings and proved that if a lattice has an SB-labelling then it is crosscut-simplicial. Some known examples of lattices with a natural SB-labelling include the join-distributive lattices, the weak order of a Coxeter group, and the Tamari lattice. Generalizing these three examples, we prove that every meet-semidistributive lattice is …
In the dynamics of a rotation of the unit circle by an irrational angle $α\in(0,1)$, we study the evolution of partitions whose atoms are finite unions of left-closed right-open intervals with endpoints lying on the past trajectory of the point $0$. Unlike the standard framework, we focus on partitions whose atoms are disconnected sets. We show that the refinements of these partitions eventually coincide with the refinements of a preimage of the Sturmian partition, which consists of two intervals $[0,1-α)$ and $[1-α,1)$. In particular, the refinements of the partitions eventually consist of co…
Using the notion of series parallel interval order, we propose a unified setting to describe Dyck lattices and Tamari lattices (two well known lattice structures on Catalan objects) in terms of basic notions of the theory of posets. As a consequence of our approach, we find an extremely simple proof of the fact that the Dyck order is a refinement of the Tamari one. Moreover, we provide a description of both the weak and the strong Bruhat order on 312-avoiding permutations, by recovering the proof of the fact that they are isomorphic to the Tamari and the Dyck order, respectively; our proof, wh…
The set of permutations on a finite set can be given the lattice structure known as the weak Bruhat order. This lattice structure is generalized to the set of words on a fixed alphabet $Σ$ = {x,y,z,...}, where each letter has a fixed number of occurrences. These lattices are known as multinomial lattices and, when card($Σ$) = 2, as lattices of lattice paths. By interpreting the letters x, y, z, . . . as axes, these words can be interpreted as discrete increasing paths on a grid of a d-dimensional cube, with d = card($Σ$).We show how to extend this ordering to images of continuous monotone func…
