A study of $m$-ary partitions whose conjugates are $q$-ary
DOI:
https://doi.org/10.13069/jacodesmath.v13i3.393Keywords:
Partitions, Conjugates, $m$-ary, CongruencesAbstract
While people have studied $m$-ary partitions of an integer $n$ and studied conjugation of partitions of $n$, these topics are rarely mixed because the $m$-ary property is almost always lost after conjugation. In a previous work, Flowers and Lockard investigated $m$-ary partitions of $n$ whose conjugates were also $m$-ary. We generalize that previous work by studying $m$-ary partitions whose conjugates are $q$-ary, where $m$ and $q$ may be distinct. We provide a family of operators on these partitions that can be used to generate all such partitions uniquely and associate a unique polynomial with each partition based on the sequence of operators used to generate it. Using the generating operators and modular arithmetic we explore many examples and families of $m$-ary partitions whose conjugates are $q$-ary.
Received: 01 July 2025 | Accepted: 07 April 2026
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Copyright (c) 2026 Geoffrey Dietz, Timothy B. Flowers, Shannon R. Lockard
This work is licensed under a Creative Commons Attribution 4.0 International License.
