Congruences modulo 9 for bipartitions withdesignated summands

Authors

ROBERT XIAOJIAN HAO
ERIN YIYING SHEN

Abstract

Andrews, Lewis, and Lovejoy studied arithmetic properties of partitions with designated summands that are defined on ordinary partitions by tagging exactly one part among parts with equal size. A bipartition of $n$ is an ordered pair of partitions $(\pi_1, \pi_2)$ with the sum of all of the parts being $n$. In this paper, we investigate arithmetic properties of bipartitions with designated summands. Let $PD_{-2}(n)$ denote the number of bipartitions of $n$ with designated summands. We establish several Ramanujan-like congruences and an infinite family of congruences modulo $9$ satisfied by $PD_{-2}(n)$.

DOI

10.3906/mat-1612-114

Keywords

Partition with designated summands, bipartition, congruence

First Page

2325

Last Page

2335

Recommended Citation

HAO, ROBERT XIAOJIAN and SHEN, ERIN YIYING (2018) "Congruences modulo 9 for bipartitions withdesignated summands," Turkish Journal of Mathematics: Vol. 42: No. 5, Article 18. https://doi.org/10.3906/mat-1612-114
Available at: https://journals.tubitak.gov.tr/math/vol42/iss5/18