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Congruences modulo powers of 5 for partition k-tuples with 5-cores

Part of: Additive number theory; partitions Combinatorics

Published online by Cambridge University Press:  17 June 2025

Xue Liang  and
Dazhao Tang Open the ORCID record for Dazhao Tang [Opens in a new window]
Xue Liang
Affiliation:
School of Mathematical Sciences, Chongqing Normal University, Chongqing, China
Dazhao Tang*
Affiliation:
School of Mathematical Sciences, Chongqing Normal University, Chongqing, China
*
Corresponding author: Dazhao Tang, email: dazhaotang@sina.com
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Abstract

A partition is called a t-core if none of its hook lengths is a multiple of t. Let $a_t(n)$ denote the number of t-core partitions of n. Garvan, Kim and Stanton proved that for any $n\geq1$ and $m\geq1$, $a_t\big(t^mn-(t^2-1)/24\big)\equiv0\pmod{t^m}$, where $t\in\{5,7,11\}$. Let $A_{t,k}(n)$ denote the number of partition k-tuples of n with t-cores. Several scholars have been subsequently investigated congruence properties modulo high powers of 5 for $A_{5,k}(n)$ with $k\in\{2,3,4\}$. In this paper, by utilizing a recurrence related to the modular equation of fifth order, we establish dozens of congruence families modulo high powers of 5 satisfied by $A_{5,k}(n)$, where $4\leq k\leq25$. Moreover, we deduce an infinite family of internal congruences modulo high powers of 5 for $A_{5,4}(n)$. In particular, we generalize greatly a recent result on a congruence family modulo high powers of 5 enjoyed by $A_{5,4}(n)$, which was proved by Saikia, Sarma and Talukdar (Indian J. Pure Appl. Math., 2024). Finally, we conjecture that there exists a similar phenomenon for $A_{5,k}(n)$ with $k\geq26$.

Keywords

partition k-tuples5-corescongruencesinternal congruencesrecurrencemodular equation

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 , First View , pp. 1 - 17
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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