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Partitions with parts separated by parity: conjugation, congruences and the mock theta functions

Part of: Combinatorics Additive number theory; partitions

Published online by Cambridge University Press:  30 November 2023

Shishuo Fu  and
Dazhao Tang Open the ORCID record for Dazhao Tang [Opens in a new window]
Shishuo Fu
Affiliation:
College of Mathematics and Statistics, Chongqing University, Huxi Campus, Chongqing 401331, People's Republic of China (fsshuo@cqu.edu.cn)
Dazhao Tang*
Affiliation:
School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, People's Republic of China (dazhaotang@sina.com)
*
*Corresponding author.
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Abstract

Noting a curious link between Andrews’ even-odd crank and the Stanley rank, we adopt a combinatorial approach building on the map of conjugation and continue the study of integer partitions with parts separated by parity. Our motivation is twofold. Firstly, we derive results for certain restricted partitions with even parts below odd parts. These include a Franklin-type involution proving a parametrized identity that generalizes Andrews’ bivariate generating function, and two families of Andrews–Beck type congruences. Secondly, we introduce several new subsets of partitions that are stable (i.e. invariant under conjugation) and explore their connections with three third-order mock theta functions $\omega (q)$, $\nu (q)$, and $\psi ^{(3)}(q)$, introduced by Ramanujan and Watson.

Keywords

partitionsconjugationAndrews–Beck type congruencesmock theta functionsparity

MSC classification

Primary: 11P81: Elementary theory of partitions
Secondary: 11P83: Partitions; congruences and congruential restrictions 05A17: Partitions of integers 05A15: Exact enumeration problems, generating functions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 21
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Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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