Weighted Forms of Euler's Theorem
William Y.C. Chen and Kathy Q. Ji
Abstract: In answer to a question of Andrews about nding 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. Pak's iterated Dyson's map and Sylvester's fish-hook bijection are the main ingredients in the weighted forms of Euler's theorem.
AMS Classification: 05A17, 11P81
Keywords: partition, rooted partition, Euler's theorem, Ramanujan's identities, Pak's iterated Dyson's map, Sylvester's fish-hook bijection