158 research outputs found

### Cambrian triangulations and their tropical realizations

This paper develops a Cambrian extension of the work of C. Ceballos, A.
Padrol and C. Sarmiento on $\nu$-Tamari lattices and their tropical
realizations. For any signature $\varepsilon \in \{\pm\}^n$, we consider a
family of $\varepsilon$-trees in bijection with the triangulations of the
$\varepsilon$-polygon. These $\varepsilon$-trees define a flag regular
triangulation $\mathcal{T}^\varepsilon$ of the subpolytope $\operatorname{conv}
\{(\mathbf{e}_{i_\bullet}, \mathbf{e}_{j_\circ}) \, | \, 0 \le i_\bullet <
j_\circ \le n+1 \}$ of the product of simplices $\triangle_{\{0_\bullet, \dots,
n_\bullet\}} \times \triangle_{\{1_\circ, \dots, (n+1)_\circ\}}$. The oriented
dual graph of the triangulation $\mathcal{T}^\varepsilon$ is the Hasse diagram
of the (type $A$) $\varepsilon$-Cambrian lattice of N. Reading. For any
$I_\bullet \subseteq \{0_\bullet, \dots, n_\bullet\}$ and $J_\circ \subseteq
\{1_\circ, \dots, (n+1)_\circ\}$, we consider the restriction
$\mathcal{T}^\varepsilon_{I_\bullet, J_\circ}$ of the triangulation
$\mathcal{T}^\varepsilon$ to the face $\triangle_{I_\bullet} \times
\triangle_{J_\circ}$. Its dual graph is naturally interpreted as the increasing
flip graph on certain $(\varepsilon, I_\bullet, J_\circ)$-trees, which is shown
to be a lattice generalizing in particular the $\nu$-Tamari lattices in the
Cambrian setting. Finally, we present an alternative geometric realization of
$\mathcal{T}^\varepsilon_{I_\bullet, J_\circ}$ as a polyhedral complex induced
by a tropical hyperplane arrangement.Comment: 16 pages, 11 figure

### Which nestohedra are removahedra?

A removahedron is a polytope obtained by deleting inequalities from the facet
description of the classical permutahedron. Relevant examples range from the
associahedra to the permutahedron itself, which raises the natural question to
characterize which nestohedra can be realized as removahedra. In this note, we
show that the nested complex of any connected building set closed under
intersection can be realized as a removahedron. We present two different
complementary proofs: one based on the building trees and the nested fan, and
the other based on Minkowski sums of dilated faces of the standard simplex. In
general, this closure condition is sufficient but not necessary to obtain
removahedra. However, we show that it is also necessary to obtain removahedra
from graphical building sets, and that it is equivalent to the corresponding
graph being chordful (i.e. any cycle induces a clique).Comment: 13 pages, 4 figures; Version 2: new Remark 2

### The greedy flip tree of a subword complex

We describe a canonical spanning tree of the ridge graph of a subword complex
on a finite Coxeter group. It is based on properties of greedy facets in
subword complexes, defined and studied in this paper. Searching this tree
yields an enumeration scheme for the facets of the subword complex. This
algorithm extends the greedy flip algorithm for pointed pseudotriangulations of
points or convex bodies in the plane.Comment: 14 pages, 10 figures; various corrections (in particular deletion of
Section 4 which contained a serious mistake pointed out by an anonymous
referee). This paper is subsumed by our joint results with Christian Stump on
"EL-labelings and canonical spanning trees for subword complexes"
(http://arxiv.org/abs/1210.1435) and will therefore not be publishe

### Brick polytopes, lattice quotients, and Hopf algebras

This paper is motivated by the interplay between the Tamari lattice, J.-L.
Loday's realization of the associahedron, and J.-L. Loday and M. Ronco's Hopf
algebra on binary trees. We show that these constructions extend in the world
of acyclic $k$-triangulations, which were already considered as the vertices of
V. Pilaud and F. Santos' brick polytopes. We describe combinatorially a natural
surjection from the permutations to the acyclic $k$-triangulations. We show
that the fibers of this surjection are the classes of the congruence $\equiv^k$
on $\mathfrak{S}_n$ defined as the transitive closure of the rewriting rule $U
ac V_1 b_1 \cdots V_k b_k W \equiv^k U ca V_1 b_1 \cdots V_k b_k W$ for letters
$a < b_1, \dots, b_k < c$ and words $U, V_1, \dots, V_k, W$ on $[n]$. We then
show that the increasing flip order on $k$-triangulations is the lattice
quotient of the weak order by this congruence. Moreover, we use this surjection
to define a Hopf subalgebra of C. Malvenuto and C. Reutenauer's Hopf algebra on
permutations, indexed by acyclic $k$-triangulations, and to describe the
product and coproduct in this algebra and its dual in term of combinatorial
operations on acyclic $k$-triangulations. Finally, we extend our results in
three directions, describing a Cambrian, a tuple, and a Schr\"oder version of
these constructions.Comment: 59 pages, 32 figure

### Geometric realizations of the accordion complex of a dissection

Consider $2n$ points on the unit circle and a reference dissection
$\mathrm{D}_\circ$ of the convex hull of the odd points. The accordion complex
of $\mathrm{D}_\circ$ is the simplicial complex of non-crossing subsets of the
diagonals with even endpoints that cross a connected subset of diagonals of
$\mathrm{D}_\circ$. In particular, this complex is an associahedron when
$\mathrm{D}_\circ$ is a triangulation and a Stokes complex when
$\mathrm{D}_\circ$ is a quadrangulation. In this paper, we provide geometric
realizations (by polytopes and fans) of the accordion complex of any reference
dissection $\mathrm{D}_\circ$, generalizing known constructions arising from
cluster algebras.Comment: 25 pages, 10 figures; Version 3: minor correction

### The brick polytope of a sorting network

The associahedron is a polytope whose graph is the graph of flips on
triangulations of a convex polygon. Pseudotriangulations and
multitriangulations generalize triangulations in two different ways, which have
been unified by Pilaud and Pocchiola in their study of flip graphs on
pseudoline arrangements with contacts supported by a given sorting network.
In this paper, we construct the brick polytope of a sorting network, obtained
as the convex hull of the brick vectors associated to each pseudoline
arrangement supported by the network. We combinatorially characterize the
vertices of this polytope, describe its faces, and decompose it as a Minkowski
sum of matroid polytopes.
Our brick polytopes include Hohlweg and Lange's many realizations of the
associahedron, which arise as brick polytopes for certain well-chosen sorting
networks. We furthermore discuss the brick polytopes of sorting networks
supporting pseudoline arrangements which correspond to multitriangulations of
convex polygons: our polytopes only realize subgraphs of the flip graphs on
multitriangulations and they cannot appear as projections of a hypothetical
multiassociahedron.Comment: 36 pages, 25 figures; Version 2 refers to the recent generalization
of our results to spherical subword complexes on finite Coxeter groups
(http://arxiv.org/abs/1111.3349

### The weak order on Weyl posets

We define a natural lattice structure on all subsets of a finite root system
that extends the weak order on the elements of the corresponding Coxeter group.
For crystallographic root systems, we show that the subposet of this lattice
induced by antisymmetric closed subsets of roots is again a lattice. We then
study further subposets of this lattice which naturally correspond to the
elements, the intervals and the faces of the permutahedron and the generalized
associahedra of the corresponding Weyl group. These results extend to arbitrary
finite crystallographic root systems the recent results of G. Chatel, V. Pilaud
and V. Pons on the weak order on posets and its induced subposets.Comment: 23 pages, 5 figure

### Graph properties of graph associahedra

A graph associahedron is a simple polytope whose face lattice encodes the
nested structure of the connected subgraphs of a given graph. In this paper, we
study certain graph properties of the 1-skeleta of graph associahedra, such as
their diameter and their Hamiltonicity. Our results extend known results for
the classical associahedra (path associahedra) and permutahedra (complete graph
associahedra). We also discuss partial extensions to the family of nestohedra.Comment: 26 pages, 20 figures. Version 2: final version with minor correction

### Multitriangulations, pseudotriangulations and primitive sorting networks

We study the set of all pseudoline arrangements with contact points which
cover a given support. We define a natural notion of flip between these
arrangements and study the graph of these flips. In particular, we provide an
enumeration algorithm for arrangements with a given support, based on the
properties of certain greedy pseudoline arrangements and on their connection
with sorting networks. Both the running time per arrangement and the working
space of our algorithm are polynomial.
As the motivation for this work, we provide in this paper a new
interpretation of both pseudotriangulations and multitriangulations in terms of
pseudoline arrangements on specific supports. This interpretation explains
their common properties and leads to a natural definition of
multipseudotriangulations, which generalizes both. We study elementary
properties of multipseudotriangulations and compare them to iterations of
pseudotriangulations.Comment: 60 pages, 40 figures; minor corrections and improvements of
presentatio

### Enumerating topological $(n_k)$-configurations

An $(n_k)$-configuration is a set of $n$ points and $n$ lines in the
projective plane such that their point-line incidence graph is $k$-regular. The
configuration is geometric, topological, or combinatorial depending on whether
lines are considered to be straight lines, pseudolines, or just combinatorial
lines. We provide an algorithm for generating, for given $n$ and $k$, all
topological $(n_k)$-configurations up to combinatorial isomorphism, without
enumerating first all combinatorial $(n_k)$-configurations. We apply this
algorithm to confirm efficiently a former result on topological
$(18_4)$-configurations, from which we obtain a new geometric
$(18_4)$-configuration. Preliminary results on $(19_4)$-configurations are also
briefly reported.Comment: 18 pages, 11 figure

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