The shape of a random numerical semigroup
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908999643824128 |
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| author | Bras-Amorós, Maria Kaplan, Nathan Singhal, Deepesh |
| author_facet | Bras-Amorós, Maria Kaplan, Nathan Singhal, Deepesh |
| contents | We study statistical properties of random numerical semigroups of a given genus. We analyze the graph of a typical numerical semigroup, understood as a function from $\mathbb{N}$ to $\mathbb{N}$. If $S$ is a numerical semigroup of genus $g$, this leads us to consider the collection of points $\left(\frac{k-1}{g-1},\frac{a_k(S)}{g} \right)$ where $1 \le k \le g$ and $a_k(S)$ denotes the $k$th smallest nonzero element of $S$. We show that as $g \rightarrow \infty$, this set of points typically becomes closer to a union of two line segments. We prove analogous results for numerical semigroups ordered by Frobenius number. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_26127 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The shape of a random numerical semigroup Bras-Amorós, Maria Kaplan, Nathan Singhal, Deepesh Combinatorics 20M14, 05A16 We study statistical properties of random numerical semigroups of a given genus. We analyze the graph of a typical numerical semigroup, understood as a function from $\mathbb{N}$ to $\mathbb{N}$. If $S$ is a numerical semigroup of genus $g$, this leads us to consider the collection of points $\left(\frac{k-1}{g-1},\frac{a_k(S)}{g} \right)$ where $1 \le k \le g$ and $a_k(S)$ denotes the $k$th smallest nonzero element of $S$. We show that as $g \rightarrow \infty$, this set of points typically becomes closer to a union of two line segments. We prove analogous results for numerical semigroups ordered by Frobenius number. |
| title | The shape of a random numerical semigroup |
| topic | Combinatorics 20M14, 05A16 |
| url | https://arxiv.org/abs/2604.26127 |