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arXiv · 2211.17090

Enumerating numerical sets associated to a numerical semigroup

April Chen·Nathan Kaplan·Liam Lawson·Christopher O'Neill·Deepesh Singhal
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Abstract

A numerical set $T$ is a subset of $\mathbb N_0$ that contains $0$ and has finite complement. The atom monoid of $T$ is the set of $x \in \mathbb N_0$ such that $x+T \subseteq T$. Marzuola and Miller introduced the anti-atom problem: how many numerical sets have a given atom monoid? This is equivalent to asking for the number of integer partitions with a given set of hook lengths. We introduce the void poset of a numerical semigroup $S$ and show that numerical sets with atom monoid $S$ are in bijection with certain order ideals of this poset. We use this characterization to answer the anti-atom problem when $S$ has small type.

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April Chen, Nathan Kaplan, Liam Lawson, Christopher O'Neill, Deepesh Singhal. 2022-11-30. Enumerating numerical sets associated to a numerical semigroup. https://arxiv.org/abs/2211.17090

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