Weighted Rogers-Ramanujan Partitions and Dyson Crank

TL;DR

This paper refines a weighted partition identity, linking partition statistics with the Dyson crank, and provides explicit generating functions for partitions grouped by these statistics.

Contribution

It introduces explicit formulas for generating functions based on partition statistics other than the norm, connecting weighted identities with the Dyson crank.

Findings

Number of partitions into even distinct parts with odd-indexed sum n equals partitions with non-negative crank.

Established explicit formulas for generating functions based on new partition statistics.

Connected weighted partition identities with the Dyson crank concept.

Abstract

In this paper we refine a weighted partition identity of Alladi. We write explicit formulas of generating functions for the number of partitions grouped with respect to a partition statistic other than the norm. We tie our weighted results and the different statistics with the crank of a partition. In particular, we prove that the number of partitions into even number of distinct parts whose odd-indexed parts' sum is n is equal to the number of partitions of n with non-negative crank.