Arithmetic Density and Congruences of t-Core Partitions

Abstract

A partition of n is called a t-core partition if none of its hook number is divisible by t. In 2019, Hirschhorn and Sellers (Bull Aust Math Soc 1:51–55, 2019) obtained a parity result for 3-core partition function $a_3(n)$. Recently, Meher and Jindal (Arithmetic density and new congruences for 3-core 590 Partitions, 2023) proved density results for $a_3(n)$, wherein we proved that $a_3(n)$ is almost always divisible by arbitrary power of 2 and 3. In this article, we prove that for a non-negative integer $\alpha ,$ $a_{3^{\alpha } m}(n)$ is almost always divisible by arbitrary power of 2 and 3. Further, we prove that $a_{t}(n)$ is almost always divisible by arbitrary power of $p_i^j,$ where j is a fixed positive integer and $t= p_1^{a_1}p_2^{a_2}\ldots p_m^{a_m}$ with primes $p_i \ge 5.$ Furthermore, by employing Radu and Seller’s approach, we obtain an algorithm and we give alternate proofs of several congruences modulo 3 and 5 for $a_{p}(n)$, where p is prime number. Our results also generalizes the results in Radu and Sellers (Acta Arith 146:43–52, 2011).