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 $p_i\ge 5$, 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.

arXiv admin note: substantial text overlap with arXiv:2404.19731