Journal of the Ramanujan Mathematical Society

Volume 41, Issue 1, March 2026 pp. 35–40.

Algebra of overpartitions

Authors: Robert Schneider
Author institution:Department of Mathematical Sciences, Michigan Technological University, Houghton, Michigan 49931, U.S.A.

Summary: 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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