Enumeration of artitions with prescribed successive rank parity blocks

Successive ranks of a partition, which were introduced by Atkin, are the difference of the $i$th row and the $i$th column in the Ferrers graph. Recently, in the study of singular overpartitions, Andrews revisited successive ranks and parity blocks. Motivated by his work, we investigate partitions with prescribed successive rank parity blocks. The main result of this paper is the generating function of partitions with exactly $d$ successive ranks and $m$ parity blocks.