Kerstetter, Franklin; O’Neill, Christopher On parametrized families of numerical semigroups. (English) Zbl 1481.20208 Commun. Algebra 48, No. 11, 4698-4717 (2020). Summary: 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 \(P_n = \langle f_1(n), \ldots, f_k(n)\rangle\) for polynomial functions \(f_i\). 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 \(P_n = \langle n, n + r_2, \ldots, n + r_k\rangle\). Our main result is to prove our conjecture in the case where each \(f_i\) 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. Cited in 1 ReviewCited in 3 Documents MSC: 20M14 Commutative semigroups 05E40 Combinatorial aspects of commutative algebra Keywords:Betti number; Frobenius number; numerical semigroup; quasipolynomial PDFBibTeX XMLCite \textit{F. Kerstetter} and \textit{C. O'Neill}, Commun. Algebra 48, No. 11, 4698--4717 (2020; Zbl 1481.20208) Full Text: DOI arXiv References: [1] Barron, T.; O’Neill, C.; Pelayo, R., On the set of elasticities in numerical monoids, Semigroup Forum, 94, 1, 37-50 (2017) · Zbl 1380.20056 [2] Bogart, T., Goodrick, J., Woods, K. Periodic behavior in families of numerical and affine semigroups via parametric Presburger arithmetic, preprint. Available at arXiv:math.CO/1911.09136. · Zbl 1404.11029 [3] Bowles, C.; Chapman, S.; Kaplan, N.; Reiser, D., On delta sets of numerical monoids, J. 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