Congruences modulo 9 for bipartitions with designated summands

Authors : Robert Xiaojian HAO, ERIN YIYING SHEN

Pages : 2325-2335

View : 16 | Download : 5

Publication Date : 0000-00-00

Article Type : Research Paper

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 $insert ignore into journalissuearticles values(\pi_1, \pi_2);$ with the sum of all of the parts being $n$. In this paper, we investigate arithmetic properties of bipartitions with designated summands. Let $PD_{-2}insert ignore into journalissuearticles values(n);$ 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 $PD_{-2}insert ignore into journalissuearticles values(n);$.
Keywords : Partition with designated summands, bipartition, congruence