Hirschhorn and Sellers studied arithmetic properties of the number of partitions with odd parts distinct. In another direction, Hammond and Lewis investigated arithmetic properties of the number of bipartitions. In this paper, we consider the number of bipartitions with odd parts distinct. Let this number be denoted by pod −2 (n). We obtain two Ramanujan-type identities for pod −2 (n), which imply that pod −2 (2n + 1) is even and pod −2 (3n + 2) is divisible by 3. Furthermore, we show that for any α ≥ 1 and n ≥ 0, pod −2 (3 2α+1 n + 23×3 2α −7 8 ) is a multiple of 3 and pod −2 (5 α+1 n + 11×5 α +1 4 ) is divisible by 5. We also find combinatorial interpretations for the two congruences modulo 2 and 3.
Abstract. Bringmann and Lovejoy introduced a rank for overpartition pairs and investigated its role in congruence properties of pp(n), the number of overpartition pairs of n. In particular, they applied the theory of Klein forms to show that there exist many Ramanujan-type congruences for the number pp(n). In this paper, we shall derive two Ramanujan-type identities and some explicit congruences for pp(n). Moreover, we find three ranks as combinatorial interpretations of the fact that pp(n) is divisible by three for any n. We also construct infinite families of congruences for pp(n) modulo 3, 5, and 9.
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