Related papers: The brick polytope of a sorting network
The polytope structure of the associahedron is decomposed into two categories, types and classes. The classification of types is related to integer partitions, whereas the classes present a new combinatorial problem. We solve this and…
A pebble tree is an ordered tree where each node receives some colored pebbles, in such a way that each unary node receives at least one pebble, and each subtree has either one more or as many leaves as pebbles of each color. We show that…
To a plane algebraic curve of degree n, Moishezon associated a braid monodromy homomorphism from a finitely generated free group to Artin's braid group B_n. Using Hansen's polynomial covering space theory, we give a new interpretation of…
Common experience suggests that many networks might possess community structure - division of vertices into groups, with a higher density of edges within groups than between them. Here we describe a new computer algorithm that detects…
Higher order networks are able to characterize data as different as functional brain networks, protein interaction networks and social networks beyond the framework of pairwise interactions. Most notably higher order networks include…
We consider the problem of computing a sequence of rankings that maximizes consumer-side utility while minimizing producer-side individual unfairness of exposure. While prior work has addressed this problem using linear or quadratic…
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…
This paper studies \emph{Dirichlet arrangements}, a generalization of graphic hyperplane arrangements arising from electrical networks and order polytopes of finite posets. We generalize descriptions of combinatorial features of graphic…
Cycle polytopes of matroids have been introduced in combinatorial optimization as a generalization of important classes of polyhedral objects like cut polytopes and Eulerian subgraph polytopes associated to graphs. Here we start an…
We define a new lattice structure on the elements of a finite Coxeter group W. This lattice, called the shard intersection order, is weaker than the weak order and has the noncrossing partition lattice NC(W) as a sublattice. The new…
In this paper, we study the connected blocks polytope, which, apart from its own merits, can be seen as the generalization of certain connectivity based or Eulerian subgraph polytopes. We provide a complete facet description of this…
We develop tools for characterizing vertices of fiber products of polytopes and apply them to simplicial distribution polytopes, a class of probability polytopes arising in quantum foundations and quantum information. In the theory of…
In this paper we examine the percolation properties of higher-order networks that have non-trivial clustering and subgraph-based assortative mixing (the tendency of vertices to connect to other vertices based on subgraph joint degree). Our…
In this paper we study polycubes: orthogonal polyhedra with axis-aligned quadrilateral faces. We present a complete characterization of polycubes of any genus based on their dual structure: a collection of oriented loops which run in each…
This dissertation investigates the geometric combinatorics of convex polytopes and connections to the behavior of the simplex method for linear programming. We focus our attention on transportation polytopes, which are sets of all tables of…
Given a graph G, we construct a simple, convex polytope whose face poset is based on the connected subgraphs of G. This provides a natural generalization of the Stasheff associahedron and the Bott-Taubes cyclohedron. Moreover, we show that…
It will be shown that the toric ring of the chain polytope of a finite partially ordered set is an algebra with straightening laws on a finite distributive lattice. Thus in particular every chain polytope possesses a regular unimodular…
A branched covering map of surfaces induces a map in the opposite direction between their arc complexes. We represent a branched covering map combinatorially using what we call a lifting picture, and use this representation to computably…
Symmetric edge polytopes are lattice polytopes associated with finite simple graphs that are of interest in both theory and applications. We investigate the facet structure of symmetric edge polytopes for various models of random graphs.…
Mirkovic and Vilonen discovered a canonical basis of algebraic cycles for the intersection homology of (the closures of the strata of) the loop Grassmannian. The moment map images of these varieties are a collection of polytopes, and they…