A Euclid style algorithm for MacMahon's partition analysis

TL;DR

This paper introduces a new Euclid-style algorithm for MacMahon's partition analysis that is easy to implement and performs well, extending polynomial-time solutions under certain conditions and applying it to generate series for magic squares of order 6.

Contribution

It presents a novel Euclid-style algorithm for MacMahon's partition analysis, combining ideas from algebraic combinatorics and computational geometry, with practical applications.

Findings

The Euclid-style algorithm is easy to implement and performs well.

A polynomial-time algorithm is generalized for fixed-dimension cases.

Generated the series for magic squares of order 6.

Abstract

Solutions to a linear Diophantine system, or lattice points in a rational convex polytope, are important concepts in algebraic combinatorics and computational geometry. The enumeration problem is fundamental and has been well studied, because it has many applications in various fields of mathematics. In algebraic combinatorics, MacMahon's partition analysis has become a general approach for linear Diophantine system related problems. Many algorithms have been developed, but "bottlenecks" always arise when dealing with complex problems. While in computational geometry, Barvinok's important result asserts the existence of a polynomial time algorithm when the dimension is fixed. However, the implementation by the LattE package of De Loera et. al. does not perform well in many situations. By combining excellent ideas in the two fields, we generalize Barvinok's result by giving a polynomial time algorithm for MacMahon's partition analysis in a suitable condition. We also present an elementary Euclid style algorithm, which might not be polynomial but is easy to implement and performs well. As applications, we contribute the generating series for magic squares of order 6.