Weighted Forms of Euler's Theorem

2005/10/06 by William Y. C. Chen, Chen, William Y. C., Kathy Q. Ji +1
Mathematics · #05A17 #11P81 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT #msc:05A17 #msc:11P81

paper · pdf · doi:10.48550/arxiv.math/0510121

14 pages

arxiv created 2005/10/06 · arxiv updated 2009/12/01

Abstract

In answer to a question of Andrews about finding combinatorial proofs of two identities in Ramanujan's "Lost" Notebook, we obtain weighted forms of Euler's theorem on partitions with odd parts and distinct parts. This work is inspired by the insight of Andrews on the connection between Ramanujan's identities and Euler's theorem. Our combinatorial formulations of Ramanujan's identities rely on the notion of rooted partitions. Iterated Dyson's map and Sylvester's bijection are the main ingredients in the weighted forms of Euler's theorem.

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