A Franklin type involution for squares

Abstract

We find an involution as a combinatorial proof of Ramanujanʼs partial theta identity. Based on this involution, we obtain a Franklin type involution on the set of partitions into distinct parts with the smallest part being odd. Compared with the involution of Bessenrodt and Pak, our involution possesses a weight-preserving property that leads to a combinatorial proof of a weighted partition theorem derived by Alladi from Ramanujanʼs partial theta identity. This gives an indirect answer to a question of Berndt, Kim and Yee. Moreover, we obtain a partition theorem based on Andrewsʼ identity and provide a combinatorial proof via certain weight assignment for our involution. A specialization of this partition theorem is related to an identity of Andrews concerning partitions into distinct nonnegative parts with the smallest part being even. Finally, we give an extension of our partition theorem which corresponds to a generalization of Andrewsʼ identity.