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ARITHMETIC PROPERTIES OF AN ANALOGUE OF t-CORE PARTITIONS
Part of:
Additive number theory; partitions
Combinatorics
Discontinuous groups and automorphic forms
Published online by Cambridge University Press: 03 June 2024
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.
MSC classification
- Type
- Research Article
- Information
- Copyright
- © The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
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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