# Infinite families of congruences for $2$ and $13$-core partitions

## TL;DR

This paper establishes infinite families of congruences and identities for 2-core and 13-core partition functions modulo 2, extending previous results and employing Hecke eigenform theory.

## Contribution

It introduces new infinite families of congruences for $a_2(n)$ and $a_{13}(n)$ modulo 2 using Hecke eigenform theory, generalizing prior work.

## Key findings

- Derived infinite families of congruences for $a_2(n)$ and $a_{13}(n)$ modulo 2.
- Established multiplicative identities for these core partition functions.
- Extended previous results by Das on similar congruences.

## 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 $a_3(n)$. Motivated by this result, both the authors \cite{MeherJindal2022} recently proved that for a non-negative integer $\alpha,$ $a_{3^{\alpha} m}(n)$ is almost always divisible by arbitrary power of $2$ and $3$ and $a_{t}(n)$ 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 $p_i \geq 5.$ In this article, by using Hecke eigenform theory, we obtain infinite families of congruences and multiplicative identities for $a_2(n)$ and $a_{13}(n)$ modulo $2$ which generalizes some results of Das \cite{Das2016}.