On the number of generalized numerical semigroups
The electronic journal of combinatorics, Tome 32 (2025) no. 3
Let $\mathsf{r}_k$ be the unique positive root of $x^k - (x+1)^{k-1} = 0$. We prove the best known bounds on the number $n_{g,d}$ of $d$-dimensional generalized numerical semigroups of genus $g$, in particular that \[n_{g,d} > C_d^{g^{(d-1)/d}} \mathsf{r}_{2^d}^g\]for some constant $C_d > 0$, 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.
DOI :
10.37236/12287
Classification :
20M14, 05A16, 11P81
Mots-clés : commutative semigroups, asymptotic enumeration, partitions, generalized numerical semigroup, GNS
Mots-clés : commutative semigroups, asymptotic enumeration, partitions, generalized numerical semigroup, GNS
Affiliations des auteurs :
Sean Li 1
@article{10_37236_12287,
author = {Sean Li},
title = {On the number of generalized numerical semigroups},
journal = {The electronic journal of combinatorics},
year = {2025},
volume = {32},
number = {3},
doi = {10.37236/12287},
zbl = {8097653},
url = {http://geodesic.mathdoc.fr/articles/10.37236/12287/}
}
Sean Li. On the number of generalized numerical semigroups. The electronic journal of combinatorics, Tome 32 (2025) no. 3. doi: 10.37236/12287
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