# Enumeration of artitions with prescribed successive rank parity blocks

## TL;DR

This paper derives the generating function for partitions characterized by a fixed number of successive ranks and parity blocks, extending the concept of successive ranks introduced by Atkin and studied by Andrews.

## Contribution

It provides a new generating function formula for partitions with prescribed successive rank parity blocks, advancing the understanding of partition structures.

## Key findings

- Derived the generating function for partitions with specified successive ranks and parity blocks.
- Extended Atkin's concept of successive ranks to include parity block constraints.
- Connected the study to recent work on singular overpartitions by Andrews.

## Abstract

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.