Ramanujanʹs lost notebook: Combinatorial proofs of identities associated with Heineʹs transformation or partial theta functions

Combinatorial proofs are given for certain entries in Ramanujanʹs lost notebook. Bijections of Sylvester, Franklin, and Wright, and applications of Algorithm Z of Zeilberger are employed. A new bijection, involving the new concept of the parity sequence of a partition, is used to prove one of Ramanujanʹs fascinating identities for a partial theta function.