Convolutive sequences, I: Through the lens of integer partition functions
Shane Chern, Dennis Eichhorn, Shishuo Fu, James A. Sellers
TL;DR
This work studies coefficient sequences of primitive eta-products that satisfy m-convolutive relations, focusing on the case m=2 with both combinatorial bijections and generating-function techniques. It provides bijective proofs for two natural 2-convolutive OEIS sequences and then extends to additional 2-convolutive sequences using analytic methods, including several eta-product decompositions and dissection identities. The paper also presents two 3-convolutive examples, develops 3-dissection identities with Ramanujan theta functions, and gives analytic proofs for these, alongside a discussion of artificially constructed convolutive sequences. It concludes with a discussion of the OEIS-search methodology, practical implications, and open questions about broader families and higher-order convolutivity.
Abstract
Motivated by the convolutive behavior of the counting function for partitions with designated summands in which all parts are odd, we consider coefficient sequences $(a_n)_{n\ge 0}$ of primitive eta-products that satisfy the generic convolutive property \begin{align*} \sum_{n\ge 0} a_{mn} q^n = \left(\sum_{n\ge 0} a_n q^n\right)^m \end{align*} for a specific positive integer $m$. Given the results of an exhaustive search of the Online Encyclopedia of Integer Sequences for such sequences for $m$ up to $6$, we first focus on the case where $m=2$ with our attention mainly paid to the combinatorics of two $2$-convolutive sequences, featuring bijective proofs for both. For other $2$-convolutive sequences discovered in the OEIS, we apply generating function manipulations to show their convolutivity. We also give two examples of $3$-convolutive sequences. Finally, we discuss other convolutive series that are not eta-products.