F50-06: Partition Analysis
There are well-known methods to find integer solutions of systems of linear equations with integer coefficients, i.e., of linear Diophantine systems of equations. A central objective of this project part is to develop new insight (in particular from discrete geometry) and new algorithmic ideas for finding non-negative integer solutions to such systems; in particular, to ones arising in combinatorial context. Partition analysis (PA) is a method to solve such problems on the level of generating functions. Despite having been invented by MacMahon more than a hundred years ago, it has been brought back to life with the help of computer algebra only recently in a joint project of G.E Andrews (PennState), the proposer and A. Riese (RISC). Within this SFB research this project should be carried on in several new directions: algorithmics, discrete geometry, and selected problems from combinatorics and number theory. Major goals to achieve are: faster algorithms for larger problem classes, new algorithmic and combinatorial connections of PA with discrete geometry (Barvinok's algorithm, etc.), generalizing PA as an operator method to new classes of multiple sums and series, and extending PA as a combinatorial investigation and proving tool (e.g., for certain kinds or variations of plane partitions).