# Arithmetic density and congruences of $t$-core partitions

## TL;DR

This paper investigates the divisibility properties of $t$-core partition functions, establishing almost always divisibility by powers of primes and providing algorithms and proofs for various congruences, extending previous results in the field.

## Contribution

It generalizes divisibility and congruence results for $t$-core partition functions, including new algorithms and proofs for prime and composite $t$, building on Radu and Seller's approach.

## Key findings

- $a_{3^{eta} m}(n)$ is almost always divisible by powers of 2 and 3.
- $a_{t}(n)$ is almost always divisible by powers of primes $p_i \\geq 5$.
- Provided algorithms and alternate proofs for congruences modulo 3 and 5.

## 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 \cite{Hirs2019} obtained a parity result for $3$-core partition function $a_3(n)$. Recently, both authors \cite{MeherJindal2022} 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 $p_i \geq 5.$ 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 \cite{radu2011a}.