  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.
