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Counting edges in factorization graphs of numerical semigroup elements

Mariah Moschetti, Christopher O'Neill

TL;DR

This work analyzes two factorization-derived graphs associated to a numerical semigroup $S$: the factorization support graph and the trade graph. It proves that the number of edges in both graphs is a quasipolynomial in $n$ for large $n$, with explicitly determined degree, period, and leading coefficient; notably, the factorization support graph has a higher degree than the trade graph. A key methodological contribution is the use of inclusion–exclusion via a poset of disjoint supports and Dual Möbius Inversion to obtain exact edge-count formulas. Additionally, the authors uncover a geometric connection by showing that a natural cubical complex built from disjoint supports is homeomorphic to real projective space $\mathbb{RP}^{k-2}$, realized via zonotopes. These results bridge combinatorial factorization structures with polyhedral and topological geometry, clarifying both computational behavior and underlying geometry.

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.

Counting edges in factorization graphs of numerical semigroup elements

TL;DR

This work analyzes two factorization-derived graphs associated to a numerical semigroup : the factorization support graph and the trade graph. It proves that the number of edges in both graphs is a quasipolynomial in for large , with explicitly determined degree, period, and leading coefficient; notably, the factorization support graph has a higher degree than the trade graph. A key methodological contribution is the use of inclusion–exclusion via a poset of disjoint supports and Dual Möbius Inversion to obtain exact edge-count formulas. Additionally, the authors uncover a geometric connection by showing that a natural cubical complex built from disjoint supports is homeomorphic to real projective space , realized via zonotopes. These results bridge combinatorial factorization structures with polyhedral and topological geometry, clarifying both computational behavior and underlying geometry.

Abstract

A numerical semigroup is an additively-closed set of non-negative integers, and a factorization of an element of is an expression of as a sum of generators of . It is known that for a given numerical semigroup , the number of factorizations of coincides with a quasipolynomial (that is, a polynomial whose coefficients are periodic functions of ). One of the standard methods for computing certain semigroup-theoretic invariants involves assembling a graph or simplicial complex derived from the factorizations of . 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 , 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.
Paper Structure (6 sections, 9 theorems, 51 equations, 4 figures)

This paper contains 6 sections, 9 theorems, 51 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background and notation
  3. Counting edges in factorization graphs
  4. Factorization support graphs
  5. Trade graphs
  6. An interesting cube complex arises

Key Result

Theorem 2.1

Let $P$ be a finite poset, and let $f: P \rightarrow \mathbb{R}$ be a function. Let the function $g:P\rightarrow \mathbb{R}$ be defined by Then

Figures (4)

  • Figure 1: The factorization support graph $\nabla_S(44)$ for $S=\langle{8,11,12}\rangle$.
  • Figure 2: The poset $\mathcal{P}_3$ and the values of $\mu(\hat{0}, -)$ in red.
  • Figure 3: Trade graphs with respect to $\rho'$ (the full graph) and $\rho$ (the green subgraph) in Example \ref{['ex:trades_example']}.
  • Figure 4: The cubical complex corresponding to the poset $(\mathcal{P}_4,\preceq)$ (left) and the zonotope of which it is a projection under Corollary \ref{['c:real_projective_space']} (right).

Theorems & Definitions (25)

  • Theorem 2.1: walkthroughcomb
  • Theorem 2.2
  • proof
  • Definition 3.1
  • Example 3.2
  • Definition 3.3
  • Example 3.4
  • Proposition 3.5
  • proof
  • Theorem 3.6
  • ...and 15 more