A generating function for all semi-magic squares and the volume of the Birkhoff polytope
J.A. De Loera
, F. Liu
and R. Yoshida
University of California Davis, Davis, CA 95616, USA
DOI: 10.1007/s10801-008-0155-y
Abstract
We present a multivariate generating function for all n\times 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\times 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.
Pages: 113–139
Keywords: keywords Birkhoff polytope; volume; lattice points; generating functions; Ehrhart polynomials
Full Text: PDF
References
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2. Barvinok, A.I.: Computing the volume, counting integral points, and exponential sums. Discrete Comput. Geom. 10, 123-141 (1993)
3. Barvinok, A.I.: A course in convexity. Graduate studies in Mathematics, vol.
54. American Math. Soc., Providence (2002)
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)
5. Beck, M., Pixton, D.: The Ehrhart polynomial of the Birkhoff polytope. Discrete Comput. Geom. 30, 623-637 (2003)
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:
7. Beck, M., Robins, S.: Computing the continuous discretely: integer-point enumeration in polyhedra. Springer undergraduate texts in Mathematics (2007)
8. Brion, M.: Points entiers dans les polyèdres convexes. Annales scientifiques de l'École Normale Supérieure Ser. 4(21), 653-663 (1988)
9. Canfield, E.R., McKay, B.: Asymptotic enumeration of integer matrices with constant row and column sums, available at math ArXiv:
10. Canfield, E.R., McKay, B.: The asymptotic volume of the Birkhoff polytope, available at math ArXi
11. Chan, C.S., Robbins, D.P.: On the volume of the polytope of doubly-stochastic matrices. Experiment.