On the number of generalized numerical semigroups

Sean Li¹

¹MIT PRIMES-USA Research Program

The electronic journal of combinatorics, Tome 32 (2025) no. 3

Résumé

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.

Détail

DOI : 10.37236/12287

Classification : 20M14, 05A16, 11P81
Mots-clés : commutative semigroups, asymptotic enumeration, partitions, generalized numerical semigroup, GNS