On the number of partitions with designated summands
William Y.C. Chen, Kathy Q. Ji, Hai-Tao Jin, and Erin Y.Y. Shen
Abstract: 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 Chan's identity on Ramanujan's cubic continued fraction and some identities on cubic theta functions. By introducing a rank for the partitions with designed summands, we give a combinatorial interpretation of the congruence for PD(3n+2).
AMS Classification: 05A17, 11P83, 05A30
Keywords: partition with designated summands, Ramanujan type identity, Ramanujan's cubic continued fraction, cubic theta function