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Title:

Hamilton Paths in Tournaments and a . . .

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An upper bound for McMullen's problem on projective transformations in R d is derived from Redei's classical theorem on Hamilton paths in tournaments.

An upper bound for McMullen's problem on projective transformations in R d is derived from Redei's classical theorem on Hamilton paths in tournaments. Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2011-12-01

Source:

http://blms.oxfordjournals.org/cgi/reprint/18/6/571.pdf

http://blms.oxfordjournals.org/cgi/reprint/18/6/571.pdf Minimize

Document Type:

text

Language:

en

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Metadata may be used without restrictions as long as the oai identifier remains attached to it.

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Title:

The Active Bijection between Regions and Simplices in Supersolvable Arrangements of Hyperplanes

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Abstract. Comparing two expressions of the Tutte polynomial of an ordered oriented matroid yields a remarkable numerical relation between the numbers of reorientations and bases with given activities. A natural activity preserving reorientation-to-basis mapping compatible with this relation is described in a series of papers by the present autho...

Abstract. Comparing two expressions of the Tutte polynomial of an ordered oriented matroid yields a remarkable numerical relation between the numbers of reorientations and bases with given activities. A natural activity preserving reorientation-to-basis mapping compatible with this relation is described in a series of papers by the present authors. This mapping, equivalent to a bijection between regions and no broken circuit subsets, provides a bijective version of several enumerative results due to Stanley, Winder, Zaslavsky, and Las Vergnas, expressing the number of acyclic orientations in graphs, or the number of regions in real arrangements of hyperplanes or pseudohyperplanes (i.e. oriented matroids), as evaluations of the Tutte polynomial. In the present paper, we consider in detail the supersolvable case – a notion introduced by Stanley – in the context of arrangements of hyperplanes. For linear orderings compatible with the supersolvable structure, special properties are available, yielding constructions significantly simpler than those in the general case. As an application, we completely carry out the computation of the active bijection for the Coxeter arrangements An and Bn. It turns out that in both cases the active bijection is closely related to a classical bijection between permutations and increasing trees. Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2008-07-01

Source:

http://www.emis.de/journals/EJC/Volume_11/PDF/v11i2r30.pdf

http://www.emis.de/journals/EJC/Volume_11/PDF/v11i2r30.pdf Minimize

Document Type:

text

Language:

en

Subjects:

matroid ; oriented matroid ; supersolvable ; Tutte polynomial ; basis ; reorientation ; region ; activity ; no broken circuit ; Coxeter arrangement ; braid arrangement ; hyperoctahedral arrangement ; bijection ; permutation ; increasing tree

matroid ; oriented matroid ; supersolvable ; Tutte polynomial ; basis ; reorientation ; region ; activity ; no broken circuit ; Coxeter arrangement ; braid arrangement ; hyperoctahedral arrangement ; bijection ; permutation ; increasing tree Minimize

DDC:

511 General principles of mathematics *(computed)*

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Metadata may be used without restrictions as long as the oai identifier remains attached to it.

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Title:

10 Points in Dimension 4 not Projectively Equivalent to the Vertices of a Convex Polytope

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Using oriented matroids, and with the help of a computer, we have found a set of 10 points in R 4 not projectively equivalent to the vertices of a convex polytope. This result confirms a conjecture of Larman [6] in dimension 4.

Using oriented matroids, and with the help of a computer, we have found a set of 10 points in R 4 not projectively equivalent to the vertices of a convex polytope. This result confirms a conjecture of Larman [6] in dimension 4. Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-08-02

Source:

http://www.lri.fr/~forge/10point.pdf

http://www.lri.fr/~forge/10point.pdf Minimize

Document Type:

text

Language:

en

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Metadata may be used without restrictions as long as the oai identifier remains attached to it.

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Devant la commission d’examen composée de:

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L’université Bordeaux I ; Directeur Inria ; Christian Krattenthaler Professeur ; Christian Krattenthaler ; Robert Cori ; Professeur Examinateur ; Michel Las ; Vergnas Directeur ; Cnrs Examinateur ; Eric Sopena ; ...

L’université Bordeaux I ; Directeur Inria ; Christian Krattenthaler Professeur ; Christian Krattenthaler ; Robert Cori ; Professeur Examinateur ; Michel Las ; Vergnas Directeur ; Cnrs Examinateur ; Eric Sopena ; Professeur Examinateur ; Emeric Gioan ; Ian Goulden ; Michel Las Vergnas ; Yvan Le Borgne ; Pierre Leroux Minimize authors

Description:

Combinatoire des cartes et polynôme de Tutte

Combinatoire des cartes et polynôme de Tutte Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2010-05-10

Source:

http://math.mit.edu/~bernardi/These.pdf

http://math.mit.edu/~bernardi/These.pdf Minimize

Document Type:

text

Language:

en

Subjects:

Philippe Flajolet. Directeur de recherche INRIA. Rapporteur

Philippe Flajolet. Directeur de recherche INRIA. Rapporteur Minimize

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Title:

The Tutte Polynomial of a Morphism of Matroids 5. Derivatives as Generating Functions of Tutte Activities

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We show that in an ordered matroid the partial derivative \partial^{p+q}t/\partialx^p\partialyq of the Tutte polynomial is p!q! times the generating function of activities of subsets with corank p and nullity q. More generally, this property holds for the 3-variable Tutte polynomial of a matroid perspective. ; Comment: 28 pages, 3 figures, 5 tables

We show that in an ordered matroid the partial derivative \partial^{p+q}t/\partialx^p\partialyq of the Tutte polynomial is p!q! times the generating function of activities of subsets with corank p and nullity q. More generally, this property holds for the 3-variable Tutte polynomial of a matroid perspective. ; Comment: 28 pages, 3 figures, 5 tables Minimize

Year of Publication:

2012-05-23

Document Type:

text

Subjects:

Mathematics - Combinatorics

Mathematics - Combinatorics Minimize

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Title:

The Tutte Polynomial of a Morphism of Matroids 6. A Multi-Faceted Counting Formula for Hyperplane Regions and Acyclic Orientations

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We show that the 4-variable generating function of certain orientation related parameters of an ordered oriented matroid is the evaluation at (x + u, y+v) of its Tutte polynomial. This evaluation contains as special cases the counting of regions in hyperplane arrangements and of acyclic orientations in graphs. Several new 2-variable expansions o...

We show that the 4-variable generating function of certain orientation related parameters of an ordered oriented matroid is the evaluation at (x + u, y+v) of its Tutte polynomial. This evaluation contains as special cases the counting of regions in hyperplane arrangements and of acyclic orientations in graphs. Several new 2-variable expansions of the Tutte polynomial of an oriented matroid follow as corollaries. This result hold more generally for oriented matroid perspectives, with specific special cases the counting of bounded regions in hyperplane arrangements or of bipolar acyclic orientations in graphs. In corollary, we obtain expressions for the partial derivatives of the Tutte polynomial as generating functions of the same orientation parameters. ; Comment: 23 pages, 2 figures, 3 tables Minimize

Year of Publication:

2012-05-24

Document Type:

text

Subjects:

Mathematics - Combinatorics

Mathematics - Combinatorics Minimize

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Title:

Hamilton Paths in Tournaments and a Problem of McMullen on Projective Transformations in Rd

Description:

An upper bound for McMullen's problem on projective transformations in R d is derived from Redei's classical theorem on Hamilton paths in tournaments.

An upper bound for McMullen's problem on projective transformations in R d is derived from Redei's classical theorem on Hamilton paths in tournaments. Minimize

Publisher:

Oxford University Press

Year of Publication:

2006-12-19

Document Type:

TEXT

Language:

en

Subjects:

NOTES AND PAPERS

NOTES AND PAPERS Minimize

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Copyright (C) 2006, London Mathematical Society

Copyright (C) 2006, London Mathematical Society Minimize

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Title:

The Tutte polynomial of a morphism of matroids I. Set-pointed matroids and matroid perspectives

Year of Publication:

1999

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Title:

The Active Bijection in Graphs, Hyperplane Arrangements, and Oriented Matroids - 1 - The fully Optimal Basis of a Bounded Region

Description:

The present paper is the first in a series of four introducing a mapping from orientations to spanning trees in graphs, from regions to simplices in real hyperplane arrangements, from reorientations to bases in oriented matroids (in order of increasing generality). We call it the active orientation-to-basis mapping, in reference to an extensive ...

The present paper is the first in a series of four introducing a mapping from orientations to spanning trees in graphs, from regions to simplices in real hyperplane arrangements, from reorientations to bases in oriented matroids (in order of increasing generality). We call it the active orientation-to-basis mapping, in reference to an extensive use of activities, a notion depending on a linear ordering, first introduced by W.T. Tutte for spanning trees in graphs. The active mapping, which preserves activities, can be considered as a bijective generalization of a polynomial identity relating two expressions - one in terms of activities of reorientations, and the other in terms of activities of bases - of the Tutte polynomial of a graph, a hyperplane arrangement or an oriented matroid. Specializations include bijective versions of well-known enumerative results related to the counting of acyclic orientations in graphs or of regions in hyperplane arrangements. Other interesting features of the active mapping are links established between linear programming and the Tutte polynomial. We consider here the bounded case of the active mapping, where bounded corresponds to bipolar orientations in the case of graphs, and refers to bounded regions in the case of real hyperplane arrangements, or of oriented matroids. In terms of activities, this is the uniactive internal case. We introduce fully optimal bases, simply defined in terms of signs, strengthening optimal bases of linear programming. An optimal basis is associated with one flat with a maximality property, whereas a fully optimal basis is equivalent to a complete flag of flats, each with a maximality property. The main results of the paper are that a bounded region has a unique fully optimal basis, and that, up to negating all signs, fully optimal bases provide a bijection between bounded regions and uniactive internal bases. In the bounded case, up to negating all signs, the active mapping is a bijection. Minimize

Year of Publication:

2009

Source:

European Journal of Combinatorics ; ISSN:0195-6698

European Journal of Combinatorics ; ISSN:0195-6698 Minimize

Document Type:

article in peer-reviewed journal

Language:

ENG

Subjects:

[INFO:INFO_DM] Computer Science/Discrete Mathematics ; AMS Classification: Primary: 52C40. Secondary: 05A99 05B35 05C20 52C35 90C05 Keywords: directed graph ; hyperplane arrangement ; matroid ; oriented matroid ; Tutte polynomial ; orientation ; basis ; activity ; acyclic orientation ; bipolar orientation ; bounded region ; bijection ; linear pr...

[INFO:INFO_DM] Computer Science/Discrete Mathematics ; AMS Classification: Primary: 52C40. Secondary: 05A99 05B35 05C20 52C35 90C05 Keywords: directed graph ; hyperplane arrangement ; matroid ; oriented matroid ; Tutte polynomial ; orientation ; basis ; activity ; acyclic orientation ; bipolar orientation ; bounded region ; bijection ; linear programming ; optimal basis Minimize

DDC:

511 General principles of mathematics *(computed)*

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Title:

A Linear Programming Construction of Fully Optimal Bases in Graphs and Hyperplane Arrangements

Description:

The fully optimal basis of a bounded acyclic oriented matroid on a linearly ordered set has been defined and studied by the present authors in a series of papers, dealing with graphs, hyperplane arrangements, and oriented matroids (in order of increasing generality). This notion provides a bijection between bipolar orientations and uniactive int...

The fully optimal basis of a bounded acyclic oriented matroid on a linearly ordered set has been defined and studied by the present authors in a series of papers, dealing with graphs, hyperplane arrangements, and oriented matroids (in order of increasing generality). This notion provides a bijection between bipolar orientations and uniactive internal spanning trees in a graph resp. bounded regions and uniactive internal bases in a hyperplane arrangement or an oriented matroid (in the sense of Tutte activities). This bijection is the basic case of a general activity preserving bijection between reorientations and subsets of an oriented matroid, called the active bijection, providing bijective versions of various classical enumerative results. Fully optimal bases can be considered as a strenghtening of optimal bases from linear programming, with a simple combinatorial definition. Our first construction used this purely combinatorial characterization, providing directly an algorithm to compute in fact the reverse bijection. A new definition uses a direct construction in terms of a linear programming. The fully optimal basis optimizes a sequence of nested faces with respect to a sequence of objective functions (whereas an optimal basis in the usual sense optimizes one vertex with respect to one objective function). This note presents this construction in terms of graphs and linear algebra. Minimize

Year of Publication:

2009

Source:

EuroComb'09: European Conference on Combinatorics, Graph Theory and Applications

EuroComb'09: European Conference on Combinatorics, Graph Theory and Applications Minimize

Document Type:

conference proceeding

Language:

ENG

Subjects:

[INFO:INFO_DM] Computer Science/Discrete Mathematics

[INFO:INFO_DM] Computer Science/Discrete Mathematics Minimize

DDC:

511 General principles of mathematics *(computed)*

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