A Franklin Type Involution for Squares

TL;DR

This paper introduces a new Franklin type involution for squares that provides combinatorial proofs for several partition identities related to Ramanujan's partial theta functions, extending classical results and answering open questions.

Contribution

It develops a novel Franklin type involution for squares, offering combinatorial proofs of multiple partition theorems and generalizations of Andrews' identities.

Findings

A Franklin type involution for squares is constructed.

Combinatorial proofs for weighted partition theorems are provided.

New interpretations for identities related to partitions with specific smallest parts are established.

Abstract

We find an involution as a combinatorial proof of a Ramanujan's partial theta identity. Based on this involution, we obtain a Franklin type involution for squares in the sense that the classical Franklin involution provides a combinatorial interpretation of Euler's pentagonal number theorem. This Franklin type involution can be considered as a solution to a problem proposed by Pak concerning the parity of the number of partitions of n into distinct parts with the smallest part being odd. Using a weighted form of our involution, we give a combinatorial proof of a weighted partition theorem derived by Alladi from Ramanujan's partial theta identity. This answers a question of Berndt, Kim and Yee. Furthermore, through a different weight assignment, we find combinatorial interpretations for another partition theorem derived by Alladi from a partial theta identity of Andrews. Moreover, we obtain a partition theorem based on Andrews' identity and provide a combinatorial proof by certain weight assignment for our involution. A specialization of our partition theorem is relate to an identity of Andrews concerning partitions into distinct nonnegative parts with the smallest part being even. Finally, we give a more general form of our partition theorem which in return corresponds to a generalization of Andrews' identity.