Basis partitions and Rogers–Ramanujan partitions Original Research Article

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 basis 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.