A numerical semigroup is an additive subsemigroup of the non-negative integers. In this article, we consider parametrized families of numerical semigroups of the form P-n = for polynomial functions fi. We conjecture that for large n, the Betti numbers, Frobenius number, genus, and type of P-n each coincide with a quasipolynomial. This conjecture has already been proven in general for Frobenius numbers, and for the remaining quantities in the special case when P-n = . Our main result is to prove our conjecture in the case where each fi is linear. In the process, we develop the notion of weighted factorization length, and generalize several known results for standard factorization lengths and delta sets to this weighted setting.
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