Journal of the
Korean Mathematical Society JKMS

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 [5] obtained a parity result for $3$-core partition function $a_3(n)$. Motivated by this result, both the authors [8] recently proved that for a non-negative integer $\alpha$, $a_{3^{\alpha} m}(n)$ is almost always divisible by an arbitrary power of $2$ and $3$ and $a_{t}(n)$ is almost always divisible by an arbitrary power of $p_i^j$, where $j$ is a fixed positive integer and $t= p_1^{a_1}p_2^{a_2}\cdots p_m^{a_m}$ 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 [2].

Keywords: $t$-core partitions, eta-quotients, congruence, modular forms

MSC numbers: Primary 11P83; Secondary 11F11