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MacMahon's Partition Analysis VI: A New Reduction Algorithm
George E.Andrews1,Peter Paule2, and Axel Riese2
1Department of Mathematics, The Pennsylvania State University, University Park, PA 16802,USA
andrews@math.psu.edu
2Research Institute for Symbolic Computation, Johannes Kepler University Linz, A--4040 Linz,Austria
{peter.paule, axel.riese}@risc.uni-linz.ac.at
Annals of Combinatorics 5 (3) p.251-270 September, 2001
AMS Subject Classification: 11P82, 68W30
Abstract:
MIn his famous book "Combinatory Analysis"MacMahon introduced Partition Analysis as a computational method for solving combinatorial problems in connection with systems of linear diophantine inequalities and equations. By developing the omega package we have shown that Partition Analysis is ideally suited for being supplemented by computer algebra methods. The object of this paper is to present a significant algorithmic improvement of this package. It overcomes a problem related to the computational treatment of roots of unity. Moreover, this new reduction strategy turns out to be superior to "The Method of Elliott" which is described in MacMahon's book. In order to make this article as self-contained as possible we give a brief introduction to Partition Analysis together with a variety of illustrative examples. For instance, the generating function of magic pentagrams is computed.
Keywords: partition analysis, magic squares, partial fraction decomposition, computer algebra

References:

1.  G.E. Andrews, P. Paule, and A. Riese, MacMahon¡¯s partition analysis III: the Omega package, Europ. J. Combin. 22 (2001) 887¨C904.

2.  G.E. Andrews and P. Paule, MacMahon¡¯s partition analysis IV: hypergeometric multisums, Sém. Lothar. Combin. B42i (1999) 24 pp.

3.  G.E. Andrews, P. Paule, and A. Riese, MacMahon¡¯s partition analysis V: bijections, recursions, and magic squares, In: Algebraic Combinatorics and Applications, A. Betten et al., Eds., Springer, 2001, pp. 1¨C39.

4.  M. Gardner, Mathematical Carnival, Alfred A. Knopf, Inc., New York, 1975.

5.  J.D. Louck, Power of a determinant with two physical applications, Int. J. Math. Math. Sci. 22 (1999) 745¨C759.

6.  P.A. MacMahon, Combinatory Analysis, 2 Vols., Cambridge University Press, Cambridge, 1915¨C1916, reprinted by Chelsea, New York, 1960.