Parametrizing numerical semigroups with multiplicity up to 5

Abstract

In this work, we give parametrizations in terms of the Kunz coordinates of numerical semigroups with multiplicity up to [Formula: see text]. We also obtain parametrizations of MED semigroups, symmetric and pseudo-symmetric numerical semigroups with multiplicity up to [Formula: see text]. These parametrizations also lead to formulas for the number of numerical semigroups, the number of MED semigroups and the number of symmetric and pseudo-symmetric numerical semigroups with multiplicity up to [Formula: see text] and given conductor.

A decomposition of Arf semigroups

“…Let λ be a partition of a positive integer n. If λ is the partition of an Arf semigroup S, then λ is called an Arf partition of n. Any positive integer n has at least one Arf partition. For example, λ = [n] is an Arf partition of n. Some of the Arf partitions of 13 are [13], [9,4], [9,3,1], [10,3], [10,2,1]. Let S be an Arf semigroup.…”

“…An Arf semigroup has maximal embedding dimension. There are several equivalent conditions on Arf semigroups, see [2,8,9,12,13].…”

The tree of numerical semigroups with low multiplicity

“…gap> List([0..9], g->Length(Filtered(NumericalSemigroupsWithGenus(g), s->Multiplicity(s)=4))); [ 0, 0, 0, 1, 3,4,6,7,9,11 ] Therefore we have the following consequence.…”

On Partitions and Arf Semigroups

“…The Arf closure of a numerical semigroup S is the smallest (with respect to set inclusion) Arf semigroup containing S. There are several equivalent conditions on Arf semigroups, see [3,8,10,14,18]. In [10,14], the authors give parametrizations of numerical semigroups with multiplicity up to . In [19], an algorithm is given for nding the Arf closure of a numerical set.…”