Arithmetic properties of an analogue of $t$-core partitions

TL;DR

This paper investigates the arithmetic densities and congruences of an analogue of $t$-core partition functions, focusing on powers of 2 and 3, and extends understanding of their divisibility properties.

Contribution

It studies the densities of the analogue $ar{a}_t(n)$ modulo powers of 2 and 3 for specific $t$, and proves new infinite families of congruences for $ar{a}_3(n)$ using Hecke operators.

Findings

Arithmetic densities of $ar{a}_t(n)$ modulo powers of 2 and 3 are characterized.

New infinite congruences for $ar{a}_3(n)$ modulo powers of 2 are established.

Results extend the understanding of divisibility properties of $t$-core analogues.

Abstract

An integer partition of a positive integer $n$ is called to be $t$-core if none of its hook lengths are divisible by $t$. Recently, Gireesh, Ray and Shivashankar [`A new analogue of $t$-core partitions', \textit{Acta Arith.} \textbf{199} (2021), 33-53] introduced an analogue $\overline{a}_t(n)$ of the $t$-core partition function $c_t(n)$. They obtained certain multiplicative formulas 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. Very recently, Bandyopadhyay and Baruah [`Arithmetic identities for some analogs of the 5-core partition function', \textit{J. Integer Seq.} \textbf{27} (2024), \# 24.4.5] proved new arithmetic identities satisfied by $\overline{a}_5(n)$. In this article, 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 on the nilpotency of Hecke operators, we prove an infinite family of congruences for $\overline{a}_3(n)$ modulo arbitrary powers of 2.