On the number of partitions with designated summands

Andrews, Lewis and Lovejoy introduced the partition function PD(n) as the number of partitions of n with designated summands, where we assume that among parts with equal size, exactly one is designated. They proved that PD(3n+2) is divisible by 3 and showed that the generating function of PD(3n) can be expressed as an infinite product of powers of (1−q2n+1) times a function F(q2). We obtain a Ramanujan type identity which implies the congruence for PD(3n+2). We also find an explicit formula for F(q2), which leads to a formula for the generating function of PD(3n). A formula for the generating function of PD(3n+1) is also obtained. Our proofs rely on Chanʼs identity on Ramanujanʼs cubic continued fraction and identities on cubic theta functions. By introducing a rank for partitions with designated summands, we give a combinatorial interpretation of the congruence for PD(3n+2).