Divisibility of an analogue of $t$-core partition function by powers of primes

Abstract: A partition of a positive integer $n$ is said to be $t$-core if none of its hook lengths are divisible by $t$. Recently, two analogues, $\overline{a}_t(n)$ and $\overline{b}_t(n)$, of the $t$-core partition function, $c_t(n)$, have been introduced by Gireesh, Ray and Shivashankar \cite{grs} and Bandyopadhyay and Baruah \cite{bb}, respectively. In this article, we prove the lacunarity of $\overline{b}_t(n)$ modulo arbitrary powers of 2 and 3 for $t=3^\alpha m$ where $\gcd(m,6)$=1. For a fixed positive integer $k$ and prime numbers $p_i\geq 5$, we also study the arithmetic density of $\overline{b}_t(n)$ modulo $p_i^k$ where $t=p_1^{a_1}\cdots p_m^{a_m}$. We further prove an infinite family of congruences for $\overline{b}_3(n)$ modulo arbitrary powers of 2 by employing a result of Ono and Taguchi on the nilpotency of Hecke operators.