Convolutive sequences, I: Through the lens of integer partition functions

TL;DR

This paper investigates special coefficient sequences arising from primitive eta-products that exhibit a convolutive property, focusing on the case m=2 with combinatorial proofs and exploring other convolutive series beyond eta-products.

Contribution

It characterizes and proves the convolutive behavior of certain eta-product sequences, especially for m=2, and extends the analysis to other convolutive series beyond eta-products.

Findings

Identified and classified 2-convolutive sequences in OEIS

Provided bijective proofs for specific 2-convolutive sequences

Presented examples of 3-convolutive sequences

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.