Bounds on the coefficients of tension and flow polynomials

Breuer, Felix; Dall, Aaron

doi:10.1007/s10801-010-0254-4

Cited by 12 publications

(6 citation statements)

References 25 publications

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“…Fortunately, there is a number of combinatorial constructs in which the inside-out polytope approach resulted in (novel) reciprocity theorems [3,7,8,9,11,12,34].…”

Section: Inside-out Polytopesmentioning

confidence: 99%

Combinatorial Reciprocity Theorems

Beck

2018

Graduate Studies in Mathematics

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A common theme of enumerative combinatorics is formed by counting functions that are polynomials evaluated at positive integers. In this expository paper, we focus on four families of such counting functions connected to hyperplane arrangements, lattice points in polyhedra, proper colorings of graphs, and P-partitions. We will see that in each instance we get interesting information out of a counting function when we evaluate it at a negative integer (and so, a priori the counting function does not make sense at this number). Our goals are to convey some of the charm these "alternative" evaluations of counting functions exhibit, and to weave a unifying thread through various combinatorial reciprocity theorems by looking at them through the lens of geometry, which will include some scenic detours through other combinatorial concepts.Date: 30 December 2011.

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“…Fortunately, there is a number of combinatorial constructs in which the inside-out polytope approach resulted in (novel) reciprocity theorems [3,7,8,9,11,12,34].…”

Section: Inside-out Polytopesmentioning

confidence: 99%

Combinatorial Reciprocity Theorems

Beck

2018

Graduate Studies in Mathematics

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“…For example, the complex in Figure 1, consisting of the boundary of the cube and the two hyperplanes, has a convex ear decomposition: Start with the boundary of the cube as triangulated by the braid arrangement, glue in the triangulated square lying on one of the hyperplanes and then glue in the two triangles on the second hyperplane one after the other. If all simplices in this complex are unimodular (as in many combinatorial applications), this leads to the following bounds, which have been successfully applied to the chromatic polynomial by Hersh and Swartz [38] and to the integral and modular flow and tension polynomials by Breuer and Dall [17].…”

Section: Coefficients Of (Quasi-)polynomialsmentioning

confidence: 99%

An Invitation to Ehrhart Theory: Polyhedral Geometry and its Applications in Enumerative Combinatorics

Breuer

2015

Lecture Notes in Computer Science

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In this expository article we give an introduction to Ehrhart theory, i.e., the theory of integer points in polyhedra, and take a tour through its applications in enumerative combinatorics. Topics include geometric modeling in combinatorics, Ehrhart's method for proving that a counting function is a polynomial, the connection between polyhedral cones, rational functions and quasisymmetric functions, methods for bounding coefficients, combinatorial reciprocity theorems, algorithms for counting integer points in polyhedra and computing rational function representations, as well as visualizations of the greatest common divisor and the Euclidean algorithm.

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“…If ∆ is an n-dimensional geometric simplicial complex in which all simplices are unimodular and if L ∆ is its Ehrhart polynomial, then f (L ∆ ) and h n (L ∆ ) are the f -and h-vectors, respectively, of the abstract simplicial complex ∆. See [9] for details.…”

Section: The Coloring Complex From the Point Of View Of Ehrhart Theorymentioning

confidence: 99%

“…The coloring complex from the point of view of Ehrhart theory. The coloring complex of an ordinary graph can be studied from the perspective of inside-out polytopes [8,9,10,11]. In this section we extend this approach to hypergraph coloring complexes.…”

Section: Ehrhart Theory the Chromatic Polynomial And Enumerative Cons...mentioning

confidence: 99%

Hypergraph coloring complexes

Breuer

Dall

Kubitzke

2012

Discrete Mathematics

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The aim of this paper is to generalize the notion of the coloring complex of a graph to hypergraphs. We present three different interpretations of those complexes–a purely combinatorial one and two geometric ones. It is shown, that most of the properties, which are known to be true for coloring complexes of graphs, break down in this more general setting, e.g., Cohen–Macaulayness and partitionability. Nevertheless, we are able to provide bounds for the f- and h-vectors of those complexes which yield new bounds on chromatic polynomials of hypergraphs. Moreover, though it is proven that the coloring complex of a hypergraph has a wedge decomposition, we provide an example showing that in general this decomposition is not homotopy equivalent to a wedge of spheres. In addition, we can completely characterize those hypergraphs whose coloring complex is connected.

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Bounds on the coefficients of tension and flow polynomials

Cited by 12 publications

References 25 publications

Combinatorial Reciprocity Theorems

Combinatorial Reciprocity Theorems

An Invitation to Ehrhart Theory: Polyhedral Geometry and its Applications in Enumerative Combinatorics

Hypergraph coloring complexes

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