On permutation numerical semigroups

TL;DR

This paper introduces the concept of n-permutation numerical semigroups, constructs multiple families for n=3 and n>2, and provides experimental data supporting a classification conjecture.

Contribution

It defines n-permutation numerical semigroups and constructs numerous examples, expanding understanding of their structure and classification.

Findings

Only three 2-permutation numerical semigroups exist.

Infinitely many n-permutation numerical semigroups for n > 2.

Experimental data supports a conjecture on classifying 3-permutation numerical semigroups.

Abstract

In this paper we introduce the notion of $n$-permutation numerical semigroup. While there are just three $2$-permutation numerical semigroups, there are infinitely many $n$-permutation numerical semigroups if $n > 2$. We construct $16$ families of $3$-permutation numerical semigroups and one family of $n$-permutation numerical semigroups. Finally we present some experimental data, which seem to support a conjecture about the classification of $3$-permutation numerical semigroups.