Counting edges in factorization graphs of numerical semigroup elements

Moschetti, Mariah; O’Neill, Christopher

doi:10.2140/involve.2025.18.861

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-4184 (online)

ISSN 1944-4176 (print)

Author index

To appear

Other MSP journals

This article is available for purchase or by subscription. See below.

Counting edges in factorization graphs of numerical semigroup elements

Mariah Moschetti and Christopher O’Neill

Vol. 18 (2025), No. 5, 861–871

DOI: 10.2140/involve.2025.18.861

Abstract

A numerical semigroup $S$ is an additively closed set of nonnegative 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 $n$ . 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.

PDF Access Denied

We have not been able to recognize your IP address 216.73.216.158 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recommendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 30.00:

Keywords

numerical semigroup, factorization, quasipolynomial

Mathematical Subject Classification

Primary: 05C25, 20M14

Milestones

Received: 12 January 2024

Revised: 4 May 2024

Accepted: 8 May 2024

Published: 20 November 2025

Communicated by Kenneth S. Berenhaut

Authors

Mariah Moschetti
Mathematics Department San Diego State University San Diego, CA United States

Christopher O’Neill
Mathematics Department San Diego State University San Diego, CA United States

Vol. 18, No. 5, 2025