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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Online-Zugang: | https://arxiv.org/abs/2403.13120 |
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| _version_ | 1866913273244286976 |
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| author | O'Dorney, Evan |
| author_facet | O'Dorney, Evan |
| contents | A \emph{numerical semigroup} is a subset $Λ$ of the nonnegative integers that is closed under addition, contains $0$, and omits only finitely many nonnegative integers (called the \emph{gaps} of $Λ$). The collection of all numerical semigroups may be visually represented by a tree of element removals, in which the children of a semigroup $Λ$ are formed by removing one element of $Λ$ that exceeds all existing gaps of $Λ$. In general, a semigroup may have many children or none at all, making it difficult to understand the number of semigroups at a given depth on the tree. We investigate the problem of estimating the number of semigroups at depth $g$ (i.e.\ of genus $g$) with $h$ children, showing that as $g$ becomes large, it tends to a proportion $ϕ^{-h-2}$ of all numerical semigroups, where $ϕ$ is the golden ratio. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_13120 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Degree asymptotics of the numerical semigroup tree O'Dorney, Evan Combinatorics 20M14 A \emph{numerical semigroup} is a subset $Λ$ of the nonnegative integers that is closed under addition, contains $0$, and omits only finitely many nonnegative integers (called the \emph{gaps} of $Λ$). The collection of all numerical semigroups may be visually represented by a tree of element removals, in which the children of a semigroup $Λ$ are formed by removing one element of $Λ$ that exceeds all existing gaps of $Λ$. In general, a semigroup may have many children or none at all, making it difficult to understand the number of semigroups at a given depth on the tree. We investigate the problem of estimating the number of semigroups at depth $g$ (i.e.\ of genus $g$) with $h$ children, showing that as $g$ becomes large, it tends to a proportion $ϕ^{-h-2}$ of all numerical semigroups, where $ϕ$ is the golden ratio. |
| title | Degree asymptotics of the numerical semigroup tree |
| topic | Combinatorics 20M14 |
| url | https://arxiv.org/abs/2403.13120 |