61 research outputs found
Lattice congruences of the weak order
We study the congruence lattice of the poset of regions of a hyperplane
arrangement, with particular emphasis on the weak order on a finite Coxeter
group. Our starting point is a theorem from a previous paper which gives a
geometric description of the poset of join-irreducibles of the congruence
lattice of the poset of regions in terms of certain polyhedral decompositions
of the hyperplanes. For a finite Coxeter system (W,S) and a subset K of S, let
\eta_K:w \mapsto w_K be the projection onto the parabolic subgroup W_K. We show
that the fibers of \eta_K constitute the smallest lattice congruence with
1\equiv s for every s\in(S-K). We give an algorithm for determining the
congruence lattice of the weak order for any finite Coxeter group and for a
finite Coxeter group of type A or B we define a directed graph on subsets or
signed subsets such that the transitive closure of the directed graph is the
poset of join-irreducibles of the congruence lattice of the weak order.Comment: 26 pages, 4 figure
Combinatorial generation via permutation languages. II. Lattice congruences
This paper deals with lattice congruences of the weak order on the symmetric
group, and initiates the investigation of the cover graphs of the corresponding
lattice quotients. These graphs also arise as the skeleta of the so-called
quotientopes, a family of polytopes recently introduced by Pilaud and Santos
[Bull. Lond. Math. Soc., 51:406-420, 2019], which generalize permutahedra,
associahedra, hypercubes and several other polytopes. We prove that all of
these graphs have a Hamilton path, which can be computed by a simple greedy
algorithm. This is an application of our framework for exhaustively generating
various classes of combinatorial objects by encoding them as permutations. We
also characterize which of these graphs are vertex-transitive or regular via
their arc diagrams, give corresponding precise and asymptotic counting results,
and we determine their minimum and maximum degrees. Moreover, we investigate
the relation between lattice congruences of the weak order and pattern-avoiding
permutations
Celebrating Loday’s associahedron
We survey Jean-Louis Loday’s vertex description of the associahedron, and its far reaching influence in combinatorics, discrete geometry, and algebra. We present in particular four topics where it plays a central role: lattice congruences of the weak order and their quotientopes, cluster algebras and their generalized associahedra, nested complexes and their nestohedra, and operads and the associahedron diagonal
From the Tamari lattice to Cambrian lattices and beyond
In this chapter, we trace the path from the Tamari lattice, via lattice
congruences of the weak order, to the definition of Cambrian lattices in the
context of finite Coxeter groups, and onward to the construction of Cambrian
fans. We then present sortable elements, the key combinatorial tool for
studying Cambrian lattices and fans. The chapter concludes with a brief
description of the applications of Cambrian lattices and sortable elements to
Coxeter-Catalan combinatorics and to cluster algebras.Comment: This is a chapter in an upcoming Tamari Festscrift. There have been
minor changes since the first version poste
Combinatorial generation via permutation languages
In this work we present a general and versatile algorithmic framework for exhaustively generating a large variety of different combinatorial objects, based on encoding them as permutations.
This approach provides a unified view on many known results and allows us to prove many new ones.
In particular, we obtain the following four classical Gray codes as special cases: the Steinhaus-Johnson-Trotter algorithm to generate all permutations of an -element set by adjacent transpositions; the binary reflected Gray code to generate all -bit strings by flipping a single bit in each step; the Gray code for generating all -vertex binary trees by rotations due to Lucas, van Baronaigien, and Ruskey; the Gray code for generating all partitions of an -element ground set by element exchanges due to Kaye.
We present two distinct applications for our new framework:
The first main application is the generation of pattern-avoiding permutations, yielding new Gray codes for different families of permutations that are characterized by the avoidance of certain classical patterns, (bi)vincular patterns, barred patterns, Bruhat-restricted patterns, mesh patterns, monotone and geometric grid classes, and many others.
We thus also obtain new Gray code algorithms for the combinatorial objects that are in bijection to these permutations, in particular for five different types of geometric rectangulations, also known as floorplans, which are divisions of a square into rectangles subject to certain restrictions.
The second main application of our framework are lattice congruences of the weak order on the symmetric group~.
Recently, Pilaud and Santos realized all those lattice congruences as -dimensional polytopes, called quotientopes, which generalize hypercubes, associahedra, permutahedra etc.
Our algorithm generates the equivalence classes of each of those lattice congruences, by producing a Hamilton path on the skeleton of the corresponding quotientope, yielding a constructive proof that each of these highly symmetric graphs is Hamiltonian.
We thus also obtain a provable notion of optimality for the Gray codes obtained from our framework: They translate into walks along the edges of a polytope
The facial weak order and its lattice quotients
We investigate a poset structure that extends the weak order on a finite
Coxeter group to the set of all faces of the permutahedron of . We call
this order the facial weak order. We first provide two alternative
characterizations of this poset: a first one, geometric, that generalizes the
notion of inversion sets of roots, and a second one, combinatorial, that uses
comparisons of the minimal and maximal length representatives of the cosets.
These characterizations are then used to show that the facial weak order is in
fact a lattice, generalizing a well-known result of A. Bj\"orner for the
classical weak order. Finally, we show that any lattice congruence of the
classical weak order induces a lattice congruence of the facial weak order, and
we give a geometric interpretation of their classes. As application, we
describe the facial boolean lattice on the faces of the cube and the facial
Cambrian lattice on the faces of the corresponding generalized associahedron.Comment: 40 pages, 13 figure
Permutrees
We introduce permutrees, a unified model for permutations, binary trees,
Cambrian trees and binary sequences. On the combinatorial side, we study the
rotation lattices on permutrees and their lattice homomorphisms, unifying the
weak order, Tamari, Cambrian and boolean lattices and the classical maps
between them. On the geometric side, we provide both the vertex and facet
descriptions of a polytope realizing the rotation lattice, specializing to the
permutahedron, the associahedra, and certain graphical zonotopes. On the
algebraic side, we construct a Hopf algebra on permutrees containing the known
Hopf algebraic structures on permutations, binary trees, Cambrian trees, and
binary sequences.Comment: 43 pages, 25 figures; Version 2: minor correction
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