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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))

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.

References

  1. Andrews, G., Paule, P., Riese, A.: MacMahon’s partition analysis VI: a new reduction algorithm. Ann. Comb. 5(3), 251–270 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  2. Monforte, A.A., Kauers, M.: Formal Laurent series in several variables. Expositiones Mathematicae 31(4), 350–367 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  3. Baldoni, V., Berline, N., De Loera, J.A., Köppe, M., Vergne, M.: How to integrate a polynomial over a simplex. Math. Comput. 80, 297–325 (2011)

    Article  MATH  Google Scholar 

  4. Barvinok, A., Woods, K.: Short rational generating functions for lattice point problems. J. Am. Math. Soc. 16(4), 957–979 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  5. Barvinok, A.I.: A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed. Math. Oper. Res. 19(4), 769–779 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  6. Barvinok, A.I.: Integer Points in Polyhedra. European Mathematical Society (2008)

    Google Scholar 

  7. Beck, M., Breuer, F., Godkin, L., Martin, J.L.: Enumerating colorings, tensions and flows in cell complexes. J. Comb. Theory, Ser. A 122, 82–106 (2014)

    Article  MathSciNet  Google Scholar 

  8. Beck, M., De Loera, J.A., Develin, M., Pfeifle, J., Stanley, R.P.: Coefficients and roots of Ehrhart polynomials. In: Barvinok, A.I. (ed.) Integer Points in Polyhedra, Proceedings of an AMS-IMS-SIAM Joint Summer Research Conference, Snowbird, Utah, 2003, pp. 1–24. AMS (2005)

    Google Scholar 

  9. Beck, M., Haase, C., Sottile, F.: Formulas of Brion, Lawrence, and Varchenko on rational generating functions for cones. Math. Intell. 31(1), 9–17 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  10. Beck, M., Robins, S.: Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra. Springer, New York (2007)

    Google Scholar 

  11. Beck, M., Sanyal, R.: Combinatorial reciprocity theorems (2014, to appear). http://math.sfsu.edu/beck/crt.html

  12. Beck, M., Zaslavsky, T.: Inside-out polytopes. Adv. Math. 205(1), 134–162 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  13. Beck, M., Zaslavsky, T.: The number of nowhere-zero flows on graphs and signed graphs. J. Comb. Theory, Ser. B 96(6), 901–918 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  14. Bergeron, N., Zabrocki, M.: The Hopf algebras of symmetric functions and quasi-symmetric functions in non-commutative variables are free and co-free. J. Algebra Appl. 8(4), 581–600 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  15. Breuer, F.: Ham sandwiches, staircases and counting polynomials. Ph.D. thesis, Freie Universität Berlin (2009)

    Google Scholar 

  16. Breuer, F.: Ehrhart \(f^*\)-coefficients of polytopal complexes are non-negative integers. Electron. J. Comb. 19(4), P16 (2012)

    MathSciNet  Google Scholar 

  17. Breuer, F., Dall, A.: Bounds on the coefficients of tension and flow polynomials. J. Algebr. Comb. 33(3), 465–482 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  18. Breuer, F., Dall, A., Kubitzke, M.: Hypergraph coloring complexes. Discrete Math. 312(16), 2407–2420 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  19. Breuer, F., Eichhorn, D., Kronholm, B.: Cranks and the geometry of combinatorial witnesses for the divisibility and periodicity of the restricted partition function (2014, in preparation)

    Google Scholar 

  20. Breuer, F., von Heymann, F.: Staircases in \(\mathbb{Z}^2\). Integers 10(6), 807–847 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  21. Breuer, F., Klivans, C.J.: Scheduling problems (2014, submitted). arXiv:1401.2978v1

  22. Breuer, F., Sanyal, R.: Ehrhart theory, modular flow reciprocity, and the Tutte polynomial. Mathematische Zeitschrift 270(1), 1–18 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  23. Breuer, F., Zafeirakopoulos, Z.: Polyhedral omega: a new algorithm for solving linear diophantine systems (2014, in preparation)

    Google Scholar 

  24. Brion, M.: Points entiers dans les polyèdres convexes. Annales scientifiques de l’École Normale Supérieure 21(4), 653–663 (1988)

    MATH  MathSciNet  Google Scholar 

  25. Bruns, W., Ichim, B., Söger, C.: The power of pyramid decomposition in normaliz (2012). arXiv:1206.1916v1

  26. Calkin, N., Wilf, H.S.: Recounting the rationals. Am. Math. Mon. 107(4), 360–363 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  27. Chari, M.K.: Two decompositions in topological combinatorics with applications to matroid complexes. Trans. Am. Math. Soc. 349(10), 3925–3943 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  28. De Loera, J.A., Hemmecke, R., Köppe, M.: Pareto optima of multicriteria integer linear programs. INFORMS J. Comput. 21(1), 39–48 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  29. De Loera, J.A., Hemmecke, R., Köppe, M.: Algebraic and geometric ideas in the theory of discrete optimization, vol. 14. SIAM (2012)

    Google Scholar 

  30. De Loera, J.A., Hemmecke, R., Tauzer, J., Yoshida, R.: Effective lattice point counting in rational convex polytopes. J. Symb. Comput. 38(4), 1273–1302 (2004)

    Article  Google Scholar 

  31. Ehrhart, E.: Sur les polyèdres rationnels homothétiques à \(n\) dimensions. C. R. Acad. Sci. Paris 254, 616–618 (1962)

    MATH  MathSciNet  Google Scholar 

  32. Fukuda, K., Prodon, A.: Double description method revisited. In: Deza, M., Euler, R., Manoussakis, I. (eds.) CCS 1995. LNCS, vol. 1120, pp. 91–111. Springer, Heidelberg (1996)

    Chapter  Google Scholar 

  33. Fukuda, K., Rosta, V.: Combinatorial face enumeration in convex polytopes. Comput. Geom. 4(4), 191–198 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  34. Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics: A Foundation for Computer Science, 2nd edn. Addison-Wesley Longman Publishing Co. Inc., Boston (1994)

    MATH  Google Scholar 

  35. Greene, C.: Acyclic orientations (notes). In: Aigner, M. (ed.) Higher Combinatorics, pp. 65–68. Reidel, Dordrecht (1977)

    Google Scholar 

  36. Haase, C., Schicho, J.: Lattice polygons and the number 2i+7. Am. Math. Mon. 116(2), 151–165 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  37. Henk, M., Tagami, M.: Lower bounds on the coefficients of Ehrhart polynomials. Eur. J. Comb. 30(1), 70–83 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  38. Hersh, P., Swartz, E.: Coloring complexes and arrangements. J. Algebr. Comb. 27(2), 205–214 (2007)

    Article  MathSciNet  Google Scholar 

  39. Jochemko, K., Sanyal, R.: Arithmetic of marked order polytopes, monotone triangle reciprocity, and partial colorings, pp. 1–16 (2013). arXiv:1206.4066v2

  40. Köppe, M., Verdoolaege, S.: Computing parametric rational generating functions with a primal Barvinok algorithm. Electron. J. Comb. 15, R16 (2008)

    Google Scholar 

  41. Lawrence, J.: Valuations and polarity. Discrete Comput. Geom. 3(1), 307–324 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  42. Lenstra, H.W.: Integer programming with a fixed number of variables. Math. Oper. Res. 8(4), 538–548 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  43. Macdonald, I.G.: Polynomials associated to finite cell complexes. J. London Math. Soc. 2(4), 181–192 (1971)

    Article  MathSciNet  Google Scholar 

  44. Pfeifle, J., Rambau, J.: Computing triangulations using oriented matroids. In: Joswig, M., Takayama, N. (eds.) Algebra, Geometry, and Software Systems, pp. 49–75. Springer, Berlin (2003)

    Chapter  Google Scholar 

  45. De Loera, J.A., Rambau, J., Santos, F.: Triangulations: Structures for Algorithms and Applications. Springer, Berlin (2010)

    Book  Google Scholar 

  46. Schrijver, A.: Theory of Linear and Integer Programming. Wiley, New York (1986)

    MATH  Google Scholar 

  47. Stanley, R.P.: Acyclic orientations of graphs. Discrete Math. 5, 171–178 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  48. Stanley, R.P.: Decompositions of rational convex polytopes. Ann. Discrete Math. 6, 333–342 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  49. Stanley, R.P.: Two poset polytopes. Discrete Comput. Geom. 1, 9–23 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  50. Stanley, R.P.: Enumerative Combinatorics. Cambridge Studies in Advanced Mathematics, vol. 2. Cambridge University Press, Cambridge (2001)

    MATH  Google Scholar 

  51. Stapledon, A.: Inequalities and Ehrhart \(\delta \)-vectors. Trans. Am. Math. Soc. 361, 5615–5626 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  52. Swartz, E.: g-Elements, finite buildings and higher Cohen-Macaulay connectivity. J. Combin. Theory, Ser. A 113, 1305–1320 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  53. Varchenko, A.N.: Combinatorics and topology of the disposition of affine hyperplanes in real space. Funct. Anal. Appl. 21(1), 9–19 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  54. Verdoolaege, S., Woods, K.: Counting with rational generating functions. J. Symb. Comput. 43(2), 75–91 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  55. Verdoolaege, S., Seghir, R., Beyls, K., Loechner, V., Bruynooghe, M.: Counting integer points in parametric polytopes using Barvinok’s rational functions. Algorithmica 48(1), 37–66 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  56. Woods, K.: Presburger arithmetic, rational generating functions, and quasi-polynomials. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013, Part II. LNCS, vol. 7966, pp. 410–421. Springer, Heidelberg (2013). http://arxiv.org/abs/1211.0020

    Chapter  Google Scholar 

  57. Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics. Springer, New York (1995)

    Book  MATH  Google Scholar 

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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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