1,513 research outputs found
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
Triangle-Free Triangulations, Hyperplane Arrangements and Shifted Tableaux
Flips of diagonals in colored triangle-free triangulations of a convex
polygon are interpreted as moves between two adjacent chambers in a certain
graphic hyperplane arrangement. Properties of geodesics in the associated flip
graph are deduced. In particular, it is shown that: (1) every diagonal is
flipped exactly once in a geodesic between distinguished pairs of antipodes;
(2) the number of geodesics between these antipodes is equal to twice the
number of Young tableaux of a truncated shifted staircase shape.Comment: figure added, plus several minor change
On properties of random dissections and triangulations
In this work we study properties of random graphs that are drawn uniformly at random from the class consisting of biconnected outerplanar graphs, or equivalently dissections of large convex polygons. We obtain very sharp concentration results for the number of vertices of any given degree, and for the number of induced copies of a given fixed graph. Our method gives similar results for random graphs from the class of triangulations of convex polygon
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