MacMahon's Partition Analysis Revisited

In his famous book ``Combinatory Analysis'' MacMahon introduced Partition Analysis (``Omega Calculus'') as a computational method for solving problems in connection with linear homogeneous diophantine inequalities and equations. The object of this talk is to illustrate that Partition Analysis is ideally suited for being supplemented by modern computer algebra methods. These new algorithms have been implemented in the form of the Mathematica package \texttt{Omega}. The talk discusses a variety of examples ranging from additive number theory to combinatorial (multi-)sums. The talk reports on joint work with George E. Andrews (PennState) and Axel Riese (RISC-Linz).