  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. We obtain a Ramanujan type identity for the
generating function of PD(3n+2) which implies the congruence of Andrews, Lewis
and Lovejoy. For PD(3n), Andrews, Lewis and Lovejoy showed that the generating
function can be expressed as an infinite product of powers of $(1-q^{2n+1})$
times a function $F(q^2)$. We find an explicit formula for $F(q^2)$, which
leads to a formula for the generating function of PD(3n). We also obtain a
formula for the generating function of PD(3n+1). 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 of Andrews,
Lewis and Lovejoy.
