Séminaire du 10 décembre 2007, Plane Partitions: MacMahon's dream came true.Peter Paule, RISC, J. Kepler University, Linz, Austria.
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.