MacMahon's Partition Analysis VI:
A New Reduction Algorithm

George E.Andrews^{1},Peter Paule^{2}, and Axel Riese^{2}

andrews@math.psu.edu

{peter.paule, axel.riese}@risc.uni-linz.ac.at

Annals of Combinatorics 5 (3) p.251-270 September, 2001

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.

References:

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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.