  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.
