Basis partitions and Rogers–Ramanujan partitions

Abstract

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=(r 1,…,r d) . Conversely, given a vector r=(r 1,…,r d) , there is a unique partition π 0 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.