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

TL;DR

This paper investigates the divisibility properties and congruences of two analogues of the $t$-core partition function, establishing lacunarity and density results modulo powers of primes, and deriving infinite families of congruences.

Contribution

It proves lacunarity of $ar{b}_t(n)$ modulo powers of 2 and 3 for specific $t$, studies the arithmetic density of $ar{b}_t(n)$ modulo prime powers, and establishes new infinite congruences for $ar{b}_3(n)$.

Findings

Lacunarity of $ar{b}_t(n)$ modulo powers of 2 and 3 for certain $t$.

Arithmetic density results of $ar{b}_t(n)$ modulo prime powers.

Infinite family of congruences for $ar{b}_3(n)$ modulo powers of 2.

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.