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

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Computer Algebra and Polynomials

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8942))

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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.

Felix Breuer was supported by Austrian Science Fund (FWF) special research group Algorithmic and Enumerative Combinatorics SFB F50-06.

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Notes

  1. 1.

    It is easy to adapt the following construction to the case of counting partitions with exactly \(m\) parts by making one inequality strict.

  2. 2.

    Classically, a polyhedral complex is a collection \(X\) of polyhedra that is closed under passing to faces, such that the intersection of any two polyhedra in \(X\) is also in \(X\) and is a face of both. In contrast, in a partial polyhedral complex some faces are allowed to be open. This means that it is possible to remove an edge from a triangle – including or excluding the incident vertices. It is sometimes useful to regard a partial polyhedral complex as subset of a fixed underlying polyhedral complex, so as to be able to refer to the vertices of the underlying complex, for example. We will disregard these technical issues in this expository paper, however.

  3. 3.

    Ehrhart formulated his theorem for polytopes, not for partial polytopal complexes. The generalization follows immediately, however, since for any partial polytopal complex \(X\) the Ehrhart function \(\mathrm{ehr }_X\) is a linear combination of Ehrhart functions of polytopes.

  4. 4.

    The affine hull of \(P\) is the smallest affine space containing \(P\). Affine spaces are the translates of linear spaces.

  5. 5.

    More precisely, we require that the affine hull of \(P\) is \(\left\{ x \; \,|\,\; A'x=b' \right\} \) and that for every row \(a\) of \(A\) the linear functional \(\left\langle a , x \right\rangle \) is not constant over \(x\in P\).

  6. 6.

    An orientation of a graph \(G\) is acyclic, if it contains no directed cycles.

  7. 7.

    Two sets \(X,Y\subset \mathbb {Z}^n\) are lattice equivalent if there exists an affine isomorphism \(x \mapsto Ax+b\) that maps \(X\) to \(Y\) and which induces a bijection on \(\mathbb {Z}^n\).

  8. 8.

    Here it is important to note that (5) works also for cones with an apex \(v\not =0\): All we have to do is take the fundamental parallelepiped \(\varPi \) to be rooted at \(v\) instead of the origin. This simply amounts to translating the fundamental parallelepiped as defined in Sect. 5 by \(v\).

  9. 9.

    A simplex \(\varDelta \) with integer vertices is unimodular if the fundamental parallelepiped of \(\mathrm{cone }(\varDelta \times \{1\})\) contains only a single integer vector: the origin. Equivalently \(\mathbb {Z}^n\cap \mathrm{cone }_\mathbb {R}(\varDelta \times \{1\}) = \mathrm{cone }_\mathbb {Z}(\varDelta \times \{1\})\).

  10. 10.

    \(M\)-vectors are defined as in Macaulay’s theorem, see for example [57, Chap. 8].

  11. 11.

    This works best if \(Q\) is the specialization of an nc-quasisymmetric function with 0-1 coefficients. Otherwise, this would require a linear combination of Ehrhart functions.

  12. 12.

    Solving a linear system of inequalities over \(\mathbb {Z}\) (or, equivalently, solving a linear system of equations over \(\mathbb {N}\)) is NP-hard. However, solving a linear system of equations over \(\mathbb {Z}\) is polynomial-time solvable, for example using the Smith normal form, see below.

  13. 13.

    We can also use the theorem of Lawrence-Varchenko to obtain an exact signed decomposition, without working modulo lines.

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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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  1. Research Institute for Symbolic Computation, Johannes Kepler University, Altenberger Str. 69, 4040, Linz, Austria

    Felix Breuer

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  1. University of Cantabria, Santander, Spain

    Jaime Gutierrez

  2. Ricam Linz, Linz, Austria

    Josef Schicho

  3. University of Caen, Caen, France

    Martin Weimann

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Breuer, F. (2015). An Invitation to Ehrhart Theory: Polyhedral Geometry and its Applications in Enumerative Combinatorics. In: Gutierrez, J., Schicho, J., Weimann, M. (eds) Computer Algebra and Polynomials. Lecture Notes in Computer Science(), vol 8942. Springer, Cham. https://doi.org/10.1007/978-3-319-15081-9_1

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