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

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 $p_i\ge 5$ 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.

To appear in Bulletin of the Australian Mathematical Society