MacMahon’s Partition Analysis V: Bijections, Recursions, and Magic Squares

A significant portion of MacMahon’s famous book “Combinatory Analysis” is devoted to the development of “Partition Analysis” as a computational method for solving problems in connection with linear homogeneous diophantine inequalities and equations, respectively. Nevertheless, MacMahon’s ideas have not received due attention with the exception of work by Richard Stanley. A long range object of a series of articles is to change this situation by demonstrating the power of MacMahon’s method in current combinatorial and partition-theoretic research. The renaissance of MacMahon’s technique partly is due to the fact that it is ideally suited for being supplemented by modern computer algebra methods. In this paper we illustrate the use of Partition Analysis and of the corresponding package Omega by focusing on three different aspects of combinatorial work: the construction of bijections (for the Refined Lecture Hall Partition Theorem), exploitation of recursive patterns (for Cayley compositions), and finding nonnegative integer solutions of linear systems of diophantine equations (for magic squares of size 3). ? Partially supported by National Science Foundation Grant DMS-9870060. ?? Supported by SFB-grant F1305 of the Austrian FWF. ? ? ? Supported by a visiting researcher grant of the J. Kepler University Linz. 2 George E. Andrews, Peter Paule, Axel Riese, and Volker Strehl