The catenary degree is an invariant that measures the distance between factorizations of elements within an atomic monoid. In this paper, we classify which finite subsets of Z≥0 occur as the set of catenary degrees of a numerical monoid (i.e., a co-finite, additive submonoid of Z≥0). In particular, we show that, with one exception, every finite subset of Z≥0 that can possibly occur as the set of catenary degrees of some atomic monoid is actually achieved by a numerical monoid.

Several recent papers have explored families of rational polyhedra whose integer points are in bijection with certain families of numerical semigroups. One such family, first introduced by Kunz, has integer points in bijection with numerical semigroups of fixed multiplicity, and another, introduced by Hellus and Waldi, has integer points corresponding to oversemigroups of numerical semigroups with two generators. In this paper, we provide a combinatorial framework from which to study both families of polyhedra. We introduce a new family of polyhedra called group cones, each constructed from some finite abelian group, from which both of the aforementioned families of polyhedra are directly determined but that are more natural to study from a standpoint of polyhedral geometry. We prove that the faces of group cones are naturally indexed by a family of finite posets, and illustrate how this combinatorial data relates to semigroups living in the corresponding faces of the other two families of polyhedra.

We present structural results on solutions to the Diophantine system $A{\boldsymbol
y} = {\boldsymbol b}$, ${\boldsymbol y} \in \mathbb Z^t_{\ge 0}$ with the smallest number
of non-zero entries. Our tools are algebraic and number theoretic in nature and include
Siegel's Lemma, generating functions, and commutative algebra. These results have some
interesting consequences in discrete optimization.