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arXiv · 2410.00941

Algebra of overpartitions

Robert Schneider
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Abstract

In a 2022 paper, Dawsey, Just and the present author prove that the set of integer partitions, taken as a monoid under a partition multiplication operation I defined in my Ph.D. work, is isomorphic to the positive integers as a monoid under integer multiplication. In this note, I extend partition multiplication to the set of overpartitions, which are of much interest in partition theory. I prove the overpartitions form an Abelian group under partition multiplication. Moreover, the overpartitions and the positive rational numbers are isomorphic as multiplicative groups. I then prove further overpartition isomorphisms and discuss approaches to a ring theory of overpartitions.

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Robert Schneider. 2024-10-01. Algebra of overpartitions. https://arxiv.org/abs/2410.00941

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