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

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 $c_i\ge 0$. 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.

Main results are unchanged, but proofs and exposition have been improved. Some details have been changed considerably including the title

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