2009/11/26 by William Y. C. Chen, Chen, William Y. C., Eric H. Liu +1
Mathematics · #05A17 #11P81 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT #msc:05A17 #msc:11P81
paper · pdf · doi:10.48550/arxiv.0911.5019
16 pages
arxiv created 2009/11/26 · openalex publication_date 2009/11/26 · arxiv updated 2009/12/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.