  In this paper, we consider the following question: "given the multiplicity
$m$ and embedding dimension $e$ of a numerical semigroup $S$, what can be said
about the cardinality $\eta$ of a minimal presentation of $S$?" We approach
this question from a combinatorial (poset-theoretic) perspective, utilizing the
recently-introduced notion of a Kunz nilsemigroup. In addition to making
significant headway on this question beyond what was previously known, in the
form of both explicit constructions and general bounds, we provide a
self-contained introduction to Kunz nilsemigroups that avoids the polyhedral
geometry necessary for much of their source material.
