Frobenius problem for semigroups $\mathsf{S}(d_{1},d_{2},d_{3})$

Abstract

We find the matrix representation of the set Δ(d3), where d3=(d1,d2,d3), of integers that are unrepresentable by d1,d2,d3 and develop a diagrammatic procedure for calculating the generating function Φ(d3;z) for the set Δ(d3). We find the Frobenius number F(d3), the genus G(d3), and the Hilbert series H(d3;z) of a graded subring for nonsymmetric and symmetric semigroups \(\mathsf{S}(\mathbf {d}^{3})\) and enhance the lower bounds of F(d3) for symmetric and nonsymmetric semigroups.