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

  • Felix Breuer
Chapter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8942)

Abstract

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.

Keywords

Polynomial Quasipolynomial Rational function Quasisymmetric function Partial polytopal complex Simplicial cone Fundamental parallelepiped Combinatorial reciprocity theorem Barvinok’s algorithm Euclidean algorithm Greatest common divisor Generating function Formal power series Integer linear programming 

Notes

Acknowledgements

I would like to thank Benjamin Nill, Peter Paule, Manuel Kauers, Christoph Koutschan and an anonymous referee for their helpful comments on earlier versions of this article. I would also like to thank Matthias Beck whose lectures and book [10] were my very own invitation to Ehrhart theory.

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Research Institute for Symbolic ComputationJohannes Kepler UniversityLinzAustria

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