Basis partitions and Rogers-Ramanujan partitions

Every partition has, for some d, a Durfee square of side d. Every partition pi 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 pi(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. (C) 1999 Elsevier Science B.V. All rights reserved.