A Franklin type involution for squares

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.

Some Combinatorial and Analytical Identities

“…In a recent paper Chen and Liu [7] generalized earlier work of Alladi [1] involving weighted partition theorems. Chen and Liu were interested in Franklin type involutions, and as an application of their technique, they proved the identity…”

Partition Identities for Ramanujan's Third-Order Mock Theta Functions

“…In fact, we can deduce two partition identities for φ(−q) and ψ(−q) analogous to Fine's identity for f (q) by employing the following partition theorem of Bessenrodt and Pak [12] which extends a theorem of Fine in [16,Theorem 5]. It is worth mentioning that there are other involutions which also imply this partition theorem, see, Berndt, Kim and Yee [11], Chen and Liu [13], and Yee [21,22]. Theorem 1.4 (Bessenrodt and Pak) Let p e do (n) (p o do (n)) denote the number of partitions of n into even (odd ) distinct parts with the smallest part being odd.…”