Counting edges in factorization graphs of numerical semigroup elements
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914162147328000 |
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| author | Moschetti, Mariah O'Neill, Christopher |
| author_facet | Moschetti, Mariah O'Neill, Christopher |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_06912 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Counting edges in factorization graphs of numerical semigroup elements Moschetti, Mariah O'Neill, Christopher Combinatorics Commutative Algebra 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. |
| title | Counting edges in factorization graphs of numerical semigroup elements |
| topic | Combinatorics Commutative Algebra |
| url | https://arxiv.org/abs/2401.06912 |