MacMahon's dream came true
by
Peter Paule
RISC, Kepler University, A-4040 Linz, Austria
Coauthors: George E. Andrews (Pennsylvania State University)

In 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. After devoting hundred pages to various aspects of Partition Analysis, he starts to consider plane partitions as a natural application domain for his method. After discussing some special cases of the full generating function for plane partitions with restricted number of rows and columns, MacMahon writes: "Our knowledge of the Omega operation is not sufficient to enable us to establish the final form of result." This talk reports on recent joint work with George E. Andrews (PennState) which shows that - despite MacMahon's negative statement - Partition Analysis indeed is powerful enough to derive the full generating function.

Date received: March 29, 2005