THE ENUMERATION PROBLEM ON NUMERICAL MONOIDS

Abstract

Even though the problem of counting points with integer coordinates on a (rational) polytope has connections to sophisticated mathematical topics like Algebraic K-Theory, Fourier-Dedekind Sums, Heegard-Floer Homology, Symplectic Geometry and more, the basic (open) problem(s) are easy to describe. For example the following has been an open problem for over 60 years: If a, b and c are coprime positive integers how many ways are there of obtaining a given natural number n as a sum of (nonnegative integer) multiples of a, b and c? The problem is giving an effective computable formula for this number f(n). We are able to find this formula for a particular case. Furthermore, we use a variety of techniques to find the secondary asymptotic in any case, along with an effective computable formula for the McNugget Monoid and a couple of infinite families.