Oriented matroid polytopes and polyhedral fans are signable

1995 ◽  
pp. 198-211 ◽  
Author(s):  
Peter Kleinschmidt ◽  
Shmuel Onn
Keyword(s):  
Oriented Matroid ◽  
Matroid Polytopes

scholarly journals On the Ehrhart Polynomial of Minimal Matroids

2021 ◽  
Author(s):  
Luis Ferroni
Keyword(s):  
Ehrhart Polynomial ◽  
Matroid Polytopes ◽  
Connected Matroid ◽  
Fixed Rank

AbstractWe provide a formula for the Ehrhart polynomial of the connected matroid of size n and rank k with the least number of bases, also known as a minimal matroid. We prove that their polytopes are Ehrhart positive and $$h^*$$ h ∗ -real-rooted (and hence unimodal). We prove that the operation of circuit-hyperplane relaxation relates minimal matroids and matroid polytopes subdivisions, and also preserves Ehrhart positivity. We state two conjectures: that indeed all matroids are $$h^*$$ h ∗ -real-rooted, and that the coefficients of the Ehrhart polynomial of a connected matroid of fixed rank and cardinality are bounded by those of the corresponding minimal matroid and the corresponding uniform matroid.

A guided tour through oriented matroid axioms

1993 ◽  
Vol 9 (2) ◽  
pp. 125-134
Author(s):  
Achim Bachem ◽  
Walter Kern
Keyword(s):  
Oriented Matroid ◽  
Guided Tour

scholarly journals Equifacetted 3-spheres as topes of nonpolytopal matroid polytopes

1995 ◽  
Vol 13 (3-4) ◽  
pp. 347-361 ◽  
Author(s):  
J. Bokowski ◽  
P. Schuchert
Keyword(s):  
Matroid Polytopes

scholarly journals Face monoid actions and tropical hyperplane arrangements

2017 ◽  
Vol 27 (08) ◽  
pp. 1001-1025
Author(s):  
Marianne Johnson ◽  
Mark Kambites
Keyword(s):  
Hyperplane Arrangement ◽  
Hyperplane Arrangements ◽  
Oriented Matroid ◽  
Definition Of ◽  
The Way

We study the combinatorics of tropical hyperplane arrangements, and their relationship to (classical) hyperplane face monoids. We show that the refinement operation on the faces of a tropical hyperplane arrangement, introduced by Ardila and Develin in their definition of a tropical oriented matroid, induces an action of the hyperplane face monoid of the classical braid arrangement on the arrangement, and hence on a number of interesting related structures. Along the way, we introduce a new characterization of the types (in the sense of Develin and Sturmfels) of points with respect to a tropical hyperplane arrangement, in terms of partial bijections which attain permanents of submatrices of a matrix which naturally encodes the arrangement.

scholarly journals On the Cocircuit Graph of an Oriented Matroid

2000 ◽  
Vol 24 (2) ◽  
pp. 257-266 ◽  
Author(s):  
R. Cordovil ◽  
K. Fukuda ◽  
A. Guedes de Oliveira
Keyword(s):  
Oriented Matroid

scholarly journals Object and image indexing based on region connection calculus and oriented matroid theory

2005 ◽  
Vol 147 (2-3) ◽  
pp. 345-361 ◽  
Author(s):  
Ernesto Staffetti ◽  
Antoni Grau ◽  
Francesc Serratosa ◽  
Alberto Sanfeliu
Keyword(s):  
Image Indexing ◽  
Oriented Matroid ◽  
Matroid Theory ◽  
Region Connection Calculus

scholarly journals Triangulations of $\Delta_{n-1} \times \Delta_{d-1}$ and Tropical Oriented Matroids

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Suho Oh ◽  
Hwanchul Yoo
Keyword(s):  
Oriented Matroids ◽  
Oriented Matroid ◽  
Nous Proposons ◽  
Combinatorial Objects ◽  
New Class ◽  
Nous Montrons ◽  
International Audience ◽  
Tropical Polytopes

International audience Develin and Sturmfels showed that regular triangulations of $\Delta_{n-1} \times \Delta_{d-1}$ can be thought of as tropical polytopes. Tropical oriented matroids were defined by Ardila and Develin, and were conjectured to be in bijection with all subdivisions of $\Delta_{n-1} \times \Delta_{d-1}$. In this paper, we show that any triangulation of $\Delta_{n-1} \times \Delta_{d-1}$ encodes a tropical oriented matroid. We also suggest a new class of combinatorial objects that may describe all subdivisions of a bigger class of polytopes. Develin et Sturmfels ont montré que les triangulations de $\Delta_{n-1} \times \Delta_{d-1}$ peuvent être considérées comme des polytopes tropicaux. Les matroïdes orientés tropicaux ont été définis par Ardila et Develin, et ils ont été conjecturés être en bijection avec les subdivisions de $\Delta_{n-1} \times \Delta_{d-1}$. Dans cet article, nous montrons que toute triangulation de $\Delta_{n-1} \times \Delta_{d-1}$ encode un matroïde orienté tropical. De plus, nous proposons une nouvelle classe d'objets combinatoires qui peuvent décrire toutes les subdivisions d'une plus grande classe de polytopes.

scholarly journals Facial structures of lattice path matroid polytopes

2020 ◽  
Vol 343 (1) ◽  
pp. 111628
Author(s):  
Suhyung An ◽  
JiYoon Jung ◽  
Sangwook Kim
Keyword(s):  
Lattice Path ◽  
Matroid Polytopes

Oriented Matroids

10.37236/25 ◽  
2002 ◽  
Vol 1000 ◽  
Author(s):  
Günter M. Ziegler
Keyword(s):  
Oriented Matroids ◽  
Entry Point ◽  
Oriented Matroid ◽  
Matroid Theory

This dynamic survey offers an “entry point” for current research in oriented matroids. For this, it provides updates on the 1993 monograph “Oriented Matroids” by Bjö̈rner, Las Vergnas, Sturmfels, White & Ziegler [85], in three parts: 1. a sketch of a few “Frontiers of Research” in oriented matroid theory, 2. an update of corrections, comments and progress as compared to [85], and 3. an extensive, complete and up-to-date bibliography of oriented matroids, comprising and extending the bibliography of [85].

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