Journal of Algebraic Combinatorics

, Volume 30, Issue 1, pp 113–139 | Cite as

A generating function for all semi-magic squares and the volume of the Birkhoff polytope



We present a multivariate generating function for all n×n nonnegative integral matrices with all row and column sums equal to a positive integer t, the so called semi-magic squares. As a consequence we obtain formulas for all coefficients of the Ehrhart polynomial of the polytope B n of n×n doubly-stochastic matrices, also known as the Birkhoff polytope. In particular we derive formulas for the volumes of B n and any of its faces.


Birkhoff polytope Volume Lattice points Generating functions Ehrhart polynomials 


  1. 1.
    Baldoni, V., De Loera, J.A, Vergne, M.: Counting integer flows in networks. Foundations of Computational Mathematics 4(3), 277–314 (2004) MATHMathSciNetGoogle Scholar
  2. 2.
    Barvinok, A.I.: Computing the volume, counting integral points, and exponential sums. Discrete Comput. Geom. 10, 123–141 (1993) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Barvinok, A.I.: A course in convexity. Graduate studies in Mathematics, vol. 54. American Math. Soc., Providence (2002) MATHGoogle Scholar
  4. 4.
    Barvinok, A.I., Pommersheim, J.: An algorithmic theory of lattice points in polyhedra. In: New Perspectives in Algebraic Combinatorics (Berkeley, CA, 1996–1997). Math. Sci. Res. Inst. Publ., vol. 38, pp. 91–147. Cambridge Univ. Press, Cambridge (1999) Google Scholar
  5. 5.
    Beck, M., Pixton, D.: The Ehrhart polynomial of the Birkhoff polytope. Discrete Comput. Geom. 30, 623–637 (2003) MATHMathSciNetGoogle Scholar
  6. 6.
    Beck, M., Hasse, C., Sottile, F.: Theorems of Brion, Lawrence, and Varchenko on rational generating functions for cones, manuscript (2007), available at math ArXiv:math.CO/0506466
  7. 7.
    Beck, M., Robins, S.: Computing the continuous discretely: integer-point enumeration in polyhedra. Springer undergraduate texts in Mathematics (2007) Google Scholar
  8. 8.
    Brion, M.: Points entiers dans les polyèdres convexes. Annales scientifiques de l’École Normale Supérieure Ser. 4(21), 653–663 (1988) MathSciNetGoogle Scholar
  9. 9.
    Canfield, E.R., McKay, B.: Asymptotic enumeration of integer matrices with constant row and column sums, available at math ArXiv:CO/0703600
  10. 10.
    Canfield, E.R., McKay, B.: The asymptotic volume of the Birkhoff polytope, available at math ArXiv:CO/0705.2422
  11. 11.
    Chan, C.S., Robbins, D.P.: On the volume of the polytope of doubly-stochastic matrices. Experiment. Math. 8(3), 291–300 (1999) MATHMathSciNetGoogle Scholar
  12. 12.
    Chan, D.P., Robbins, C.S, Yuen, D.S: On the volume of a certain polytope. Experiment. Math. 9(1), 91–99 (2000) MATHMathSciNetGoogle Scholar
  13. 13.
    De Loera, J.A., Hemmecke, R., Tauzer, J., Yoshida, R.: Effective Lattice Point Counting in Rational Convex Polytopes. Journal of Symbolic Computation 38, 1273–1302 (2004) CrossRefMathSciNetGoogle Scholar
  14. 14.
    De Loera, J.A, Rambau, J., Santos, F.: Triangulations: Structures and Algorithms. Manuscript (2008) Google Scholar
  15. 15.
    Diaconis, P., Gangolli, A.: Rectangular Arrays with Fixed Margins. IMA Series on Volumes in Mathematics and its Applications, vol. 72, pp. 15–41. Springer, Berlin (1995) Google Scholar
  16. 16.
    Ehrhart, E.: Polynômes Arithmétiques et Méthode des Polyédres en Combinatoire. Birkhauser, Basel (1977) MATHGoogle Scholar
  17. 17.
    Filliman, P.: The volume of duals and sections of polytopes. Mathematika 39, 67–80 (1992) MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Fulton, W.: Introduction to Toric Varieties. Princeton University Press, Princeton (1993). 180 pages MATHGoogle Scholar
  19. 19.
    Kuperberg, G.: A generalization of Filliman duality. Proceedings of the AMS 131(12), 3893–3899 (2003) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Lawrence, J.: Polytope volume computation. Math. Comput. 57, 259–271 (1991) MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Schrijver, A.: Theory of Linear and Integer Programming. Wiley, New York (1986) MATHGoogle Scholar
  22. 22.
    Stanley, R.P.: Enumerative Combinatorics, 2nd ed., vol. I. Cambridge University Press, Cambridge (1997) MATHGoogle Scholar
  23. 23.
    Sturmfels, B.: Gröbner Bases and Convex Polytopes. University Lecture Series, vol. 8. AMS, Providence (1995) Google Scholar
  24. 24.
    Yemelichev, V.A., Kovalev, M.M., Kratsov, M.K.: Polytopes, Graphs and Optimisation. Cambridge Univ. Press, Cambridge (1984) MATHGoogle Scholar
  25. 25.
    Zeilberger, D.: Proof of a conjecture of Chan, Robbins, and Yuen. Electronic Transactions on Numerical Analysis 9, 147–148 (1999) MATHMathSciNetGoogle Scholar
  26. 26.
    Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer, New York (1995). 370 pages MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.University of California DavisDavisUSA

Personalised recommendations