On the parity of the number of partitions with odd multiplicities
| Autor: | Fabrizio Zanello, James A. Sellers |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Algebra and Number Theory
Mathematics - Number Theory Mathematics::Number Theory Primary: 11P83 Secondary: 05A17 11P84 11E25 Computer Science::Information Retrieval Astrophysics::Instrumentation and Methods for Astrophysics Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) Multiplicity (mathematics) Combinatorics FOS: Mathematics Mathematics - Combinatorics Computer Science::General Literature Combinatorics (math.CO) Number Theory (math.NT) Parity (mathematics) Mathematics |
| Zdroj: | International Journal of Number Theory. 17:1717-1728 |
| ISSN: | 1793-7310 1793-0421 |
| DOI: | 10.1142/s1793042121500573 |
| Popis: | Recently, Hirschhorn and the first author considered the parity of the function $a(n)$ which counts the number of integer partitions of $n$ wherein each part appears with odd multiplicity. They derived an effective characterization of the parity of $a(2m)$ based solely on properties of $m.$ In this note, we quickly reprove their result, and then extend it to an explicit characterization of the parity of $a(n)$ for all $n\not\equiv 7 \pmod{8}.$ We also exhibit some infinite families of congruences modulo 2 which follow from these characterizations. We conclude by discussing the case $n\equiv 7 \pmod{8}$, where, interestingly, the behavior of $a(n)$ modulo 2 appears to be entirely different. In particular, we conjecture that, asymptotically, $a(8m+7)$ is odd precisely $50\%$ of the time. This conjecture, whose broad generalization to the context of eta-quotients will be the topic of a subsequent paper, remains wide open. Comment: Minor revisions. To appear in the Int. J. Number Theory |
| Databáze: | OpenAIRE |
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