Weighted Rogers–Ramanujan partitions and Dyson crank
In this paper, we refine a weighted partition identity of Alladi. We write formulas for generating functions for the number of partitions grouped with respect to a partition statistic other than the norm. We tie our weighted results as well as 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.
KeywordsDyson crank Partitions Rogers–Ramanujan Weighted partition identities
Mathematics Subject Classification05A15 05A17 05A19 11B75 11P81 11P84
The author would like to thank George E. Andrews and Alexander Berkovich for their guidance. We would like to express gratitude to the anonymous referee for the constructive comments and making this work more presentable. The author would also like to thank Alexander Berkovich, Jeramiah Hocutt, Frank Patane, and John Pfeilsticker for their helpful comments on the manuscript.
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