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ARITHMETIC PROPERTIES OF AN ANALOGUE OF t-CORE PARTITIONS

Part of: Additive number theory; partitions Discontinuous groups and automorphic forms Combinatorics

Published online by Cambridge University Press:  03 June 2024

PRANJAL TALUKDAR Open the ORCID record for PRANJAL TALUKDAR [Opens in a new window]
PRANJAL TALUKDAR*
Affiliation:
Department of Mathematical Sciences, Tezpur University, Napaam 784028, Assam, India
*
e-mail: [email protected]
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Abstract

An integer partition of a positive integer n is called t-core if none of its hook lengths is divisible by t. Gireesh et al. [‘A new analogue of t-core partitions’, Acta Arith. 199 (2021), 33–53] introduced an analogue $\overline {a}_t(n)$ of the t-core partition function. They obtained multiplicative formulae and arithmetic identities for $\overline {a}_t(n)$ where $t \in \{3,4,5,8\}$ and studied the arithmetic density of $\overline {a}_t(n)$ modulo $p_i^{\,j}$ where $t=p_1^{a_1}\cdots p_m^{a_m}$ and $p_i\geq 5$ are primes. Bandyopadhyay and Baruah [‘Arithmetic identities for some analogs of the 5-core partition function’, J. Integer Seq. 27 (2024), Article no. 24.4.5] proved new arithmetic identities satisfied by $\overline {a}_5(n)$. We study the arithmetic densities of $\overline {a}_t(n)$ modulo arbitrary powers of 2 and 3 for $t=3^\alpha m$ where $\gcd (m,6)$=1. Also, employing a result of Ono and Taguchi [‘2-adic properties of certain modular forms and their applications to arithmetic functions’, Int. J. Number Theory 1 (2005), 75–101] on the nilpotency of Hecke operators, we prove an infinite family of congruences for $\overline {a}_3(n)$ modulo arbitrary powers of 2.

Keywords

t-core partitionanalogue of t-core partitiontheta functionmodular formarithmetic density

MSC classification

Primary: 11P83: Partitions; congruences and congruential restrictions
Secondary: 05A17: Partitions of integers 11F11: Holomorphic modular forms of integral weight
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 111 , Issue 1 , February 2025 , pp. 68 - 79
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The author was partially supported by the Council of Scientific and Industrial Research (CSIR), Government of India, under the CSIR-JRF scheme (Grant No. 09/0796(12991)/2021-EMR-I).

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