Weighted Rogers-Ramanujan Partitions and Dyson Crank

2016/03/01 by Uncu, Ali Kemal
#05A15 #05A17 #05A19 #11B75 #11P81 #11P84 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1603.00399

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.

Related

• Partitions weighted by the parity of the crank
• New Weighted Rogers-Ramanujan Partition Theorems and their Implications
• Variation on a theme of Nathan Fine. New weighted partition identities
• New Weighted Partition Theorems with the Emphasis on the Smallest Part of Partitions
• Non-Rascoe partitions and a rank parity function associated to the Rogers-Ramanujan partitions