Counting edges in factorization graphs of numerical semigroup elements

Abstract

A numerical semigroup $S$ is an additively-closed set of non-negative integers, and a factorization of an element $n$ of $S$ is an expression of $n$ as a sum of generators of $S$. It is known that for a given numerical semigroup $S$, the number of factorizations of $n$ coincides with a quasipolynomial (that is, a polynomial whose coefficients are periodic functions of $n$). One of the standard methods for computing certain semigroup-theoretic invariants involves assembling a graph or simplicial complex derived from the factorizations of $n$. In this paper, we prove that for two such graphs (which we call the factorization support graph and the trade graph), the number of edges coincides with a quasipolynomial function of $n$, and identify the degree, period, and leading coefficient of each. In the process, we uncover a surprising geometric connection: a combinatorially-assembled cubical complex that is homeomorphic to real projective space.