Maximal Denumerant of a Numerical Semigroup With Embedding Dimension Less Than Four

TL;DR

This paper investigates the maximum number of maximal factorizations in numerical semigroups with embedding dimension less than four, providing formulas and bounds for this quantity.

Contribution

It introduces the concept of the maximal denumerant and derives explicit formulas for semigroups with embedding dimension less than four.

Findings

The number of maximal factorizations is always bounded.

Formulas for the maximal denumerant are established for embedding dimension less than four.

The maximal denumerant can be explicitly computed in these cases.

Abstract

Given a numerical semigroup $S = < a_1, a_2,..., a_t>$ and $s\in S$, we consider the factorization $s = c_1 a_1 + c_2 a_2 +... + c_t a_t$ where $c_i\ge0$. Such a factorization is {\em maximal} if $c_1+c_2+...+c_t$ is a maximum over all such factorizations of $s$. We show that the number of maximal factorizations, varying over the elements in $S$, is always bounded. Thus, we define $\dx(S)$ to be the maximum number of maximal factorizations of elements in $S$. We study maximal factorizations in depth when $S$ has embedding dimension less than four, and establish formulas for $\dx(S)$ in this case.