Related papers: Combinatorial flag arrangements
Structured matrices with symbolic sizes appear frequently in the literature, especially in the description of algorithms for linear algebra. Recent work has treated these symbolic structured matrices themselves as computational objects,…
It is known that there exist hyperplane arrangements with same underlying matroid that admit non-homotopy equivalent complement manifolds. In this work we show that, in any rank, complex central hyperplane arrangements with up to 7…
We introduce a combinatorial characterization of simpliciality for arrangements of hyperplanes. We then give a sharp upper bound for the number of hyperplanes of such an arrangement in the projective plane over a finite field, and present…
We study a generalization of the concept of harmonic conjugation from projective geometry and full algebraic matroids to a larger class of matroids called \emph{harmonic matroids}. We use harmonic conjugation to construct a projective plane…
In this paper we develop the theory of cyclic flats of $q$-matroids. We show that the lattice of cyclic flats, together with their ranks, uniquely determines a $q$-matroid and hence derive a new $q$-cryptomorphism. We introduce the notion…
In this note, we propose a simple-looking but broad conjecture about star-algebras over the field of real numbers. The conjecture enables many matrix decompositions to be represented by star-algebras and star-ideals. This paper is written…
We consider three forms of composition of matroids, each of which extends the category of bimatroids to a rigid monoidal category. Many well-known constructions are functorial or defined by morphisms in these categories. Motivating examples…
This thesis proposes a framework based on a notion of combinatorial cell complex (cc) whose cells are defined simply as finite sets of vertices. The cells of a cc are subject to four axioms involving a rank function that assigns a rank (or…
We study valuated matroids, their tropical incidence relations, flag matroids and total positivity. This leads to a characterization of permutahedral subdivisions, namely subdivisions of regular permutahedra into generalized permutahedra.…
In this paper we introduce the notion of hybrid trigonometric parametrization as a tuple of real rational expressions involving circular and hyperbolic trigonometric functions as well as monomials, with the restriction that variables in…
We give a general notion of combinatory completeness with respect to a faithful cartesian club and use it systematically to obtain characterisations of a number of different kinds of applicative system. Each faithful cartesian club…
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…
A binary frame template is a device for creating binary matroids from graphic or cographic matroids. Such matroids are said to conform or coconform to the template. We introduce a preorder on these templates and determine the nontrivial…
We define and prove isomorphisms between three combinatorial classes involving labeled trees. We also give an alternative proof by means of generating functions.
We introduce a notion of duality (due to Brylawski) that generalizes matroid duality to arbitrary rank functions. This generalized duality allows for generalized operations (deletion and contraction) and a generalized polynomial based on…
We introduce combinatorial objects which are parameterized by the positive part of the tropical Grassmannian $Gr(k,n)$. Our method is to relate the Grassmannian to configuration spaces of flags. By work of the first author, and of Goncharov…
We prove the existence of complexified real arrangements with the same combinatorics but different embeddings in the complex projective plane. Such pair of arrangements has an additional property: they admit conjugated equations on the ring…
We develop a combinatorial model of the associated Hermite polynomials and their moments, and prove their orthogonality with a sign-reversing involution. We find combinatorial interpretations of the moments as complete matchings, connected…
Hyperplane arrangements form the latest addition to the zoo of combinatorial objects dealt with by polymake. We report on their implementation and on a algorithm to compute the associated cell decomposition. The implemented algorithm…
We study paving matroids, their realization spaces, and their closures, along with matroid varieties and circuit varieties. Within this context, we introduce three distinct methods for generating polynomials within the associated ideals of…