In this paper, we investigate arithmetic properties of bipartitions with designated summands. Let P D−2(n) denote the number of bipartitions of n with designated summands. | Turk J Math (2018) 42: 2325 – 2335 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article Congruences modulo 9 for bipartitions with designated summands 1 Robert Xiaojian HAO1 ,, Erin Yiying SHEN2,∗, Department of Mathematics and Physics, Nanjing Institute of Technology, Nanjing, . China 2 School of Science, Hohai University, Nanjing, . China Received: • Accepted/Published Online: • Final Version: 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 (π1 , π2 ) with the sum of all of the parts being n . In this paper, we investigate arithmetic properties of bipartitions with designated summands. Let P D−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 P D−2 (n) . Key words: Partition with designated summands, bipartition, congruence 1. Introduction In [1], Andrews et al. investigated the number of partitions with designated summands that are defined on ordinary partitions by designating exactly one part of each part size. Let P D(n) denote the number of partitions of n with designated summands. For instance, there are ten partitions of 4 with designated summands: 4′ , 3′ + 1′ , 2′ + 2, 2 + 2′ , 2′ + 1′ + 1, 2′ + 1 + 1′ , 1′ + 1 + 1 + 1, 1 + 1′ + 1 + 1, 1 + 1 + 1′ + 1, 1 + 1 + 1 + 1′ . Thus, P D(4) = 10 . Andrews et al. [1] obtained the generating function of P D(n) as given by ∞ ∑ n=0 P D(n)q n = (q 6 ; q 6 )∞ f6 = , (q; q)∞ (q 2 ; q 2 )∞ (q 3 ; q 3 )∞ f1 f2 f3 () where here and throughout this paper (a; q)∞ stands for the q -shifted factorial (a; q)∞ = ∞ ∏ (1 − aq n−1 ), |q| < 1, n=1 and for any positive .
