On the Number of Generalized Numerical Semigroups

Let rₖ be the unique positive root of xᵏ - (x+1)ᵏ⁻¹ = 0. We prove the best known bounds on the number ng,d of d-dimensional generalized numerical semigroups of genus g, in particular that \[ng,d > C_d^{g(d-1)/d} r2^d^g\]for some constant Cd > 0, which can be made explicit. To do this, we extend the notion of multiplicity and depth to generalized numerical semigroups and show our lower bound is sharp for semigroups of depth 2. We also show other bounds on special classes of semigroups by introducing partition labelings, which extend the notion of Kunz words to the general setting.