Multiparameter counting of numerical semigroups by Frobenius number, genus, and multiplicity
Abstract
A numerical semigroup is a subset of the nonnegative integers, closed under addition and with finite complement. The size of the complement is its genus. The problem of counting semigroups by the largest gap got very important through the so-called Frobenius problem, first documented in 1884. Accordingly, the largest gap is called the Frobenius number. In the last two decades, counting by the genus has become a subject of even more intense study, mostly because of the non-solved conjectures on its monotonic and super-Fibonacci growth. It is known that the number of semigroups of each genus grows asymptotically with the genus like the Fibonacci numbers and that the number of semigroups of each Frobenius number grows asymptotically in such a way that each number doubles the previous but one. We prove a formula for the number of numerical semigroups of each Frobenius number, genus, and multiplicity (first nonzero nongap), under the assumption that the multiplicity m and the Frobenius number F satisfy m≥(F+1)/3. It has been proved that asymptotically almost all semigroups satisfy this condition. From this formula we prove multiparameter recurrence relations. They give, under the required restriction, a multiparameter exact version of the increasing behaviours previously mentioned. Indeed, we prove that the number nF,m,g of semigroups with Frobenius number F, multiplicity m and genus g satisfies nF,m,g=nF−2,m−1,g−1+nF−2,m−1,g−2 and that the number nF,m of semigroups with Frobenius number F and multiplicity m satisfies nF,m=2nF−2,m−1.
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BibTeX
@misc{brasamoros2026multiparameter,
title = {{Multiparameter counting of numerical semigroups by Frobenius number, genus, and multiplicity}},
author = {Maria Bras-Amorós and Vicenç Torra},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.11559},
eprint = {2611.11559},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.11559},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Multiparameter counting of numerical semigroups by Frobenius number, genus, and multiplicity
AU - Maria Bras-Amorós
AU - Vicenç Torra
PY - 2026
DA - 2026/11/08
PB - arXiv
DO - 10.48550/arxiv.2611.11559
UR - https://arxiv.org/abs/2611.11559
AB - A numerical semigroup is a subset of the nonnegative integers, closed under addition and with finite complement. The size of the complement is its genus. The problem of counting semigroups by the largest gap got very important through the so-called Frobenius problem, first documented in 1884. Accordingly, the largest gap is called the Frobenius number. In the last two decades, counting by the genus has become a subject of even more intense study, mostly because of the non-solved conjectures on its monotonic and super-Fibonacci growth. It is known that the number of semigroups of each genus grows asymptotically with the genus like the Fibonacci numbers and that the number of semigroups of each Frobenius number grows asymptotically in such a way that each number doubles the previous but one. We prove a formula for the number of numerical semigroups of each Frobenius number, genus, and multiplicity (first nonzero nongap), under the assumption that the multiplicity m and the Frobenius number F satisfy m≥(F+1)/3. It has been proved that asymptotically almost all semigroups satisfy this condition. From this formula we prove multiparameter recurrence relations. They give, under the required restriction, a multiparameter exact version of the increasing behaviours previously mentioned. Indeed, we prove that the number n_F,m,g of semigroups with Frobenius number F, multiplicity m and genus g satisfies n_F,m,g=n_F−2,m−1,g−1+n_F−2,m−1,g−2 and that the number n_F,m of semigroups with Frobenius number F and multiplicity m satisfies n_F,m=2n_F−2,m−1.
ER -