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

TL;DR

This paper introduces Ehrhart theory, exploring how polyhedral geometry aids in counting integer points, with applications in combinatorics, including polynomial counting, reciprocity, and computational algorithms.

Contribution

It provides an accessible overview of Ehrhart theory and its diverse applications in enumerative combinatorics, highlighting new connections and computational methods.

Findings

Ehrhart's method proves counting functions are polynomials.

Connections between polyhedral cones, rational functions, and quasisymmetric functions are established.

Algorithms for counting integer points and computing rational functions are discussed.

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 couting 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.