Basis partitions and Rogers-Ramanujan partitions

Every partition has, for some d, a Durfee square of side d. Every partition π with Durfee square of side d gives rise to a “successive rank vector” r = (r1, · · · , rd). Conversely, given a vector r = (r1, · · · , rd), there is a unique partition π0 of minimal size called the basis partition with r as its successive rank vector. We give a quick derivation of the generating function for b(n, d), the number of basic partitions of n with Durfee square side d, and show that b(n, d) is a weighted sum over all Rogers-Ramanujan partitions of n into d parts.