Congruences modulo 9 for bipartitions withdesignated summands

Authors

ROBERT XIAOJIAN HAO
ERIN YIYING SHEN

DOI

10.3906/mat-1612-114

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 $PD_{-2}(n)$.

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