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

Abstract

Given a numerical semigroup $S =$ 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.Comment: Main results are unchanged, but proofs and exposition have been improved. Some details have been changed considerably including the titl