Weighted forms of Eulerʹs theorem

In answer to a question of Andrews about finding combinatorial proofs of two identities in Ramanujanʹs “lost” notebook, we obtain weighted forms of Eulerʹs theorem on partitions with odd parts and distinct parts. This work is inspired by the insight of Andrews on the connection between Ramanujanʹs identities and Eulerʹs theorem. Our combinatorial formulations of Ramanujanʹs identities rely on the notion of rooted partitions. Pakʹs iterated Dysonʹs map and Sylvesterʹs fish-hook bijection are the main ingredients in the weighted forms of Eulerʹs theorem.