Convolutive sequences, I: Through the lens of integer partition functions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908692425736192 |
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| author | Chern, Shane Eichhorn, Dennis Fu, Shishuo Sellers, James A. |
| author_facet | Chern, Shane Eichhorn, Dennis Fu, Shishuo Sellers, James A. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_10965 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convolutive sequences, I: Through the lens of integer partition functions Chern, Shane Eichhorn, Dennis Fu, Shishuo Sellers, James A. Combinatorics Discrete Mathematics Number Theory 05A17, 11P81 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. |
| title | Convolutive sequences, I: Through the lens of integer partition functions |
| topic | Combinatorics Discrete Mathematics Number Theory 05A17, 11P81 |
| url | https://arxiv.org/abs/2507.10965 |