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Arithmetic properties of Schur-type overpartitions

Isaac A. Broudy and Jeremy Lovejoy

Vol. 15 (2022), No. 3, 489–505
Abstract

We investigate arithmetic properties of an overpartition counting function that first arose in connection with Schur’s partition theorem and a universal mock theta function. Motivated by work of Basil Gordon on the Rogers–Ramanujan identities, we first give a complete characterization of the parity of this overpartition function in the progressions 2n + 1, 4n + 2, and 8n + 4 in terms of the factorization of An + B for certain A and B. We then find similar characterizations of the residue modulo 4 in the progressions 8n + 5 and 8n + 7. Finally, we prove some Ramanujan-type congruences modulo 5. Our proofs use basic facts about modular forms and some elementary algebraic number theory.

Keywords
partitions, overpartitions, parity, eta-quotients, binary quadratic forms
Mathematical Subject Classification
Primary: 11P83
Secondary: 11E25, 11P84
Milestones
Received: 2 August 2021
Revised: 16 November 2021
Accepted: 18 November 2021
Published: 2 December 2022

Communicated by Ken Ono
Authors
Isaac A. Broudy
Department of Mathematics
University of California
Berkeley, CA
United States
Jeremy Lovejoy
CNRS
Université de Paris
Paris
France