Arithmetic properties of bipartitions with designated summands

Abstract

In 2002 Andrews, Lewis and Lovejoy introduced partition function P D(n), the number of partitions of n with designated summands and using modular forms they obtained many congruences modulo 3 and powers of 2. For example, they proved P D(3n + 2) ≡ 0 (mod 3). In this paper, we study various arithmetic properties of P D 2 (n) modulo 3 and powers of 2, where P D 2 (n) denotes the number of bipartitions of n with designated summands. We obtain congruences like P D 2 (3 α+3 (3n + 2)) ≡ 0 (mod 3), P D 2 (3 α+3 (6n + 4)) ≡ 0 (mod 3), P D 2 (24n + 15) ≡ 0 (mod 2 5 ), P D 2 (24n+23) ≡ 0 (mod 2 5 ), P D 2 (24n+12) ≡ 0 (mod 12) and P D 2 (18n+15) ≡ 0 (mod 48).

Congruences modulo 9 for bipartitions with designated summands

“…In a very recent work, Naika and Shivashankar [18] investigated the arithmetic properties of the generating function of P D −2 (n) . They proved that this function satisfied various congruence properties modulo 3 and powers of 2.…”