Basis partitions and Rogers-Ramanujan partitions (Q1301850)

From the paper's abstract: 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' \( {\mathbf r} = (r_1, \dots, r_d)\). Conversely, given a vector \({\mathbf r} = (r_1, \dots, r_d)\), there is a unique partition \(\pi_0\) of minimal size called the basis partition with \({\mathbf r}\) 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.