Infinite families of congruences for $2$ and $13$-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 \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 $t=p_1^{a_1}{p}_2^{a_2}\dots p_m^{a_m}$ with primes $p_i\ge 5.$ 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}.

arXiv admin note: substantial text overlap with arXiv:2302.11830