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

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, āₜ(n) and b̄ₜ(n), of the t-core partition function, cₜ(n), have been introduced by Gireesh, Ray and Shivashankar {grs} and Bandyopadhyay and Baruah {bb}, respectively. In this article, we prove the lacunarity of b̄ₜ(n) modulo arbitrary powers of 2 and 3 for t=3^α m where (m,6)=1. For a fixed positive integer k and prime numbers pᵢ≥ 5, we also study the arithmetic density of b̄ₜ(n) modulo pᵢᵏ where t=p₁a₁⋯ pₘaₘ. We further prove an infinite family of congruences for b̄₃(n) modulo arbitrary powers of 2 by employing a result of Ono and Taguchi on the nilpotency of Hecke operators.