Advertisement

Polyhedral Cones of Magic Cubes and Squares

  • Maya Ahmed
  • Jesús De Loera
  • Raymond Hemmecke
Part of the Algorithms and Combinatorics book series (AC, volume 25)

Abstract

Using computational algebraic geometry techniques and Hilbert bases of polyhedral cones we derive explicit formulas and generating functions for the number of magic squares and magic cubes.

Keywords

Hilbert Series Hilbert Function Monomial Ideal Polyhedral Cone Semigroup Ring 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alvis, D. and Kinyon, M. Birkhoff’s theorem for Panstochastic matrices, Amer. Math. Monthly, (2001), Vol 108, no.1, 28–37.MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Andrews, W.S. Magic squares and cubes, second edition, Dover Publications, Inc., New York, N.Y. 1960.Google Scholar
  3. 3.
    Beck, M. and Pixton D. The Ehrhart polynomial of the Birkhoff Polytope e-print: arXiv:math.CO/0202267, (2002)Google Scholar
  4. 4.
    Beck, M., Cohen, M., Cuomo, J., and Gribelyuk, P. The number of “magic squares” and hypercubes, e-print: arXiv:math.CO/0201013, (2002).Google Scholar
  5. 5.
    Bigatti, A.M Computation of Hilbert-Poincaré Series, J. Pure Appl. Algebra, 119/3, (1997), 237–253.MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bona, M., Sur l’enumeration des cubes magiques, C.R.Acad.Sci.Paris Ser.I Math. 316 (1993), no.7, 633–636.MathSciNetMATHGoogle Scholar
  7. 7.
    Bruns, W. and Koch, R. NORMALIZ, computing normalizations of affine semigroups, Available via anonymous ftp from ftp//ftp.mathematik.unionabrueck. de/pub/osm/kommalg/software/Google Scholar
  8. 8.
    Capani, A., Niesi, G., and Robbiano, L., CoCoA, a system for doing Computations in Commutative Algebra, Available via anonymous ftp from cocoa.dima.unige.it, (2000).Google Scholar
  9. 9.
    Chan, C.S. and Robbins, D.P. On the volume of the polytope of doubly stochastic matrices, Experimental Math. Vol 8, No. 3, (1999), 291–300.MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Contejean, E. and Devie, H. Resolution de systemes lineaires d’equations diophantienes C. R. Acad. Sci. Paris, 313: 115–120, 1991. Serie IMathSciNetMATHGoogle Scholar
  11. 11.
    Cox, D., Little, J., and O’Shea, D. Ideals, varieties, and Algorithms, Springer Verlag, Undergraduate Text, 2nd Edition, 1997.Google Scholar
  12. 12.
    De Loera, J.A. and Sturmfels B. Algebraic unimodular counting to appear in Mathematical Programming.Google Scholar
  13. 13.
    Ehrhart, E.Sur un probléme de géométrie diophantienne linéaire II,J. Reine Angew. Math. 227 (1967), 25–49.MathSciNetGoogle Scholar
  14. 14.
    Ehrhart, E.Figures magiques et methode des polyedres J.Reine Angew.Math299/300(1978), 51–63.MathSciNetGoogle Scholar
  15. 15.
    Ehrhart, E. Sur les carrés magiques, C.R. Acad. Sci. Paris 227 A, (1973), 575–577.Google Scholar
  16. 16.
    Halleck, E.Q. Magic squares subclasses as linear Diophantine systems, Ph.D. dissertation, Univ. of California San Diego, 2000, 187 pages.Google Scholar
  17. 17.
    Henk, M. and Weismantel, R. On Hilbert bases of polyhedral cones, Results in Mathematics, 32, (1997), 298–303.MathSciNetMATHGoogle Scholar
  18. 18.
    Hemmecke, R. On the Computation of Hilbert Bases of Cones, in: “Mathematical Software, ICMS 2002”, A.M. Cohen, X.-S. Gao, N. Takayama, eds., World Scientific, 2002. Software implementation 4ti2 available from http://www.4ti2.de.
  19. Gardner, M. Martin Gardner’s New mathematical Diversions from Scientific American, Simon and Schuster, New York 1966. pp 162–172Google Scholar
  20. 20.
    MacMahon, P.A. Combinatorial Analysis, Chelsea, 1960volumes I and II reprint of 1917 edition.Google Scholar
  21. 21.
    Pasles, P.C. The lost squares of Dr. Franklin, Amer. Math. Monthly, 108, (2001), no. 6, 489–511.MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Pottier, L. Bornes et algorithme de calcul des générateurs des solutions de systémes diophantiens linéaires, C. R. Acad. Sci. Paris, 311, (1990) no. 12, 813–816.MathSciNetMATHGoogle Scholar
  23. 23.
    Pottier, L. Minimal solutions of linear Diophantine systems: bounds and algorithms, In Rewriting techniques and applications (Como, 1991), 162–173, Lecture Notes in Comput. Sci., 488, Springer, Berlin, 1991.Google Scholar
  24. 24.
    Schrijver, A. Theory of Linear and Integer Programming. Wiley-Interscience, 1986.Google Scholar
  25. 25.
    Stanley, R.P. Linear homogeneous diophantine equations and magic labellings of graphs, Duke Math J. 40 (1973), 607–632.MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Stanley, R.P. Combinatorics and commutative algebra, Progress in Mathematics, 41, Birkhaüser Boston, MA, 1983.Google Scholar
  27. 27.
    Stanley, R.P. Enumerative Combinatorics, Volume I, Cambridge, 1997.CrossRefMATHGoogle Scholar
  28. 28.
    Sturmfels, B. Gröbner bases and convex polytopes, university lecture series, vol. 8, AMS, Providence RI, (1996).Google Scholar
  29. 29.
    Vergne M. and Baldoni-Silva W. Residues formulae for volumes and Ehrhart polynomials of convex polytopes. manuscript 81 pages. available at math.ArXiv, CO/0103097.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Maya Ahmed
    • 1
  • Jesús De Loera
    • 1
  • Raymond Hemmecke
    • 1
  1. 1.Dept. of MathematicsUniversity of CaliforniaDavisUSA

Personalised recommendations