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Some congruences for 12-coloured generalized Frobenius partitions

Part of: Additive number theory; partitions Combinatorics

Published online by Cambridge University Press:  02 May 2024

Su-Ping Cui ,
Nancy S. S. Gu  and
Dazhao Tang Open the ORCID record for Dazhao Tang [Opens in a new window]
Su-Ping Cui
Affiliation:
School of Mathematical Sciences, Qufu Normal University, Qufu, PR China
Nancy S. S. Gu
Affiliation:
Center for Combinatorics, LPMC, Nankai University, Tianjin, PR China
Dazhao Tang*
Affiliation:
School of Mathematical Sciences, Chongqing Normal University, Chongqing, PR China
*
Corresponding author: Dazhao Tang, email: dazhaotang@sina.com
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Abstract

In his 1984 AMS Memoir, Andrews introduced the family of functions $c\phi_k(n)$, the number of k-coloured generalized Frobenius partitions of n. In 2019, Chan, Wang and Yang systematically studied the arithmetic properties of $\textrm{C}\Phi_k(q)$ for $2\leq k\leq17$ by utilizing the theory of modular forms, where $\textrm{C}\Phi_k(q)$ denotes the generating function of $c\phi_k(n)$. In this paper, we first establish another expression of $\textrm{C}\Phi_{12}(q)$ with integer coefficients, then prove some congruences modulo small powers of 3 for $c\phi_{12}(n)$ by using some parameterized identities of theta functions due to A. Alaca, S. Alaca and Williams. Finally, we conjecture three families of congruences modulo powers of 3 satisfied by $c\phi_{12}(n)$.

Keywords

congruencesgeneralized Frobenius partitionsgenerating functionsinteger matrix exact covering systemsparameterized identities

MSC classification

Primary: 11P83: Partitions; congruences and congruential restrictions
Secondary: 05A17: Partitions of integers 05A15: Exact enumeration problems, generating functions
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 67 , Issue 3 , August 2024 , pp. 778 - 793
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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