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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2506.10222 |
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| _version_ | 1866915850829692928 |
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| author | Cyrusian, Sogol Kaplan, Nathan |
| author_facet | Cyrusian, Sogol Kaplan, Nathan |
| contents | There has been significant recent interest in studying how the number of numerical semigroups of genus $g$ behaves as a function of $g$. Bras-Amorós has shown how to organize the collection of numerical semigroups of genus $g$ into a rooted tree called the ordinarization tree. The ordinarization number of a numerical semigroup $S$ is the length of the path from $S$ back to the root of the tree. We study the problem of counting numerical semigroups of genus $g$ with a fixed ordinarization number $r$. We show how this can be interpreted as a counting problem about integer points in a certain rational polyhedral cone and use ideas from Ehrhart theory to study this problem. We give a formula for the number of numerical semigroups of genus $g$ and ordinarization number $2$, building on the corresponding result of Bras-Amorós for ordinarization number $1$. We show that the ordinarization number of a numerical semigroup generated by two elements is equal to the number of integer points in a certain right triangle with rational vertices. We consider the analogous problem for supersymmetric numerical semigroups with more generators. We also study ordinarization numbers of numerical semigroups generated by an interval. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_10222 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ordinarization numbers of numerical semigroups Cyrusian, Sogol Kaplan, Nathan Combinatorics There has been significant recent interest in studying how the number of numerical semigroups of genus $g$ behaves as a function of $g$. Bras-Amorós has shown how to organize the collection of numerical semigroups of genus $g$ into a rooted tree called the ordinarization tree. The ordinarization number of a numerical semigroup $S$ is the length of the path from $S$ back to the root of the tree. We study the problem of counting numerical semigroups of genus $g$ with a fixed ordinarization number $r$. We show how this can be interpreted as a counting problem about integer points in a certain rational polyhedral cone and use ideas from Ehrhart theory to study this problem. We give a formula for the number of numerical semigroups of genus $g$ and ordinarization number $2$, building on the corresponding result of Bras-Amorós for ordinarization number $1$. We show that the ordinarization number of a numerical semigroup generated by two elements is equal to the number of integer points in a certain right triangle with rational vertices. We consider the analogous problem for supersymmetric numerical semigroups with more generators. We also study ordinarization numbers of numerical semigroups generated by an interval. |
| title | Ordinarization numbers of numerical semigroups |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2506.10222 |