Arithmetic properties of Schur-type overpartitions

Broudy, Isaac A.; Lovejoy, Jeremy

doi:10.2140/involve.2022.15.489

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Arithmetic properties of Schur-type overpartitions

Isaac A. Broudy and Jeremy Lovejoy

Vol. 15 (2022), No. 3, 489–505

DOI: 10.2140/involve.2022.15.489

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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>2</mn><mi>n</mi> <mo class="MathClass-bin">+</mo> <mn>1</mn></math>$ , $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>4</mn><mi>n</mi> <mo class="MathClass-bin">+</mo> <mn>2</mn></math>$ , and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>8</mn><mi>n</mi> <mo class="MathClass-bin">+</mo> <mn>4</mn></math>$ in terms of the factorization of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>A</mi><mi>n</mi> <mo class="MathClass-bin">+</mo> <mi>B</mi></math>$ for certain $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>A</mi></math>$ and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi></math>$ . We then find similar characterizations of the residue modulo $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>4</mn></math>$ in the progressions $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>8</mn><mi>n</mi> <mo class="MathClass-bin">+</mo> <mn>5</mn></math>$ and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>8</mn><mi>n</mi> <mo class="MathClass-bin">+</mo> <mn>7</mn></math>$ . Finally, we prove some Ramanujan-type congruences modulo $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>5</mn></math>$ . 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

Vol. 15, No. 3, 2022